File Almost integer in triangle.svg thumb 250px Ed Pegg, Jr. noted that the length math d frac sqrt 1842630 690 sqrt 5831385 60 math that is very close to 7 7.0000000857367483286 ca. ref name MathWorld In recreational mathematics an almost integer is any number that is very close to an integer . Well known examples of almost integers are high powers of the golden ratio math phi frac 1 sqrt5 2 approx 1.618 , math , for example math phi 17 frac 3571 1597 sqrt5 2 approx 3571.00028 , math math phi 18 2889 1292 sqrt5 approx 5777.999827 , math math phi 19 frac 9349 4181 sqrt5 2 approx 9349.000107 , math The fact that these powers approach integers is non coincidental, which is trivially seen because the golden ratio is a Pisot Vijayaraghavan number . Other occurrences of non coincidental near integers involve the three largest Heegner number s math e pi sqrt 43 approx 884736743.999777466 , math math e pi sqrt 67 approx 147197952743.999998662454 , math math e pi sqrt 163 approx 262537412640768743.99999999999925007 , math where the non coincidence can be better appreciated when expressed in the common simple form ref http groups.google.com.ph group sci.math.research browse thread thread 3d24137c9a860893?hl en ref math e pi sqrt 43 12 3 9 2 1 3 744 2.225 cdots times 10 4 , math math e pi sqrt 67 12 3 21 2 1 3 744 1.337 cdots times 10 6 , math math e pi sqrt 163 12 3 231 2 1 3 744 7.499 cdots times 10 13 , math where math 21 3 times7,231 3 times7 times11,744 24 times 31 , math and the reason for the squares being due to certain Eisenstein series . The constant math e pi sqrt 163 , math is sometimes referred to as Ramanujan s constant . Almost integers involving the mathematical constants pi and E mathematical constant e have often puzzled mathematicians. An example is math e pi pi 19.999099979189 cdots , math To date, no explanation has been given for why Gelfond s constant math ... AlmostInteger.html Almost Integer at MathWorld ref which is therefore regarded to be a mathematical ... more details
In mathematics, a polyphase sequence is a sequence whose terms are complex number complex root of unity roots of unity math a n e i frac 2 pi q x n , math where x sub n sub is an integer . Polyphase sequences is an important class of sequences and play important roles in synchronizing sequence design. References Pingzhi Fan and Michael Darnell, Sequence Design for Communications Applications , 1996 Category Sequences and series ... more details
. The M bius transform is also a nonlinear transformation, only possible for integersequence ...In mathematics , a sequence transformation is an Operator mathematics operator acting on a given space of sequence s. Sequence transformations include linear mappings such as convolution with another sequence, and resummation of a sequence and, more generally, are commonly used for series acceleration , that is, for improving the rate of convergence of a slowly convergent sequence or series mathematics series . Sequence transformations are also commonly used to compute the antilimit of a divergent ... for sequence transformations include the binomial transform , M bius transform , Stirling transform and others. Definitions For a given sequence math S s n n in N , , math the transformed sequence is math mathbf T S S s n n in N , , math where the members of the transformed sequence are usually computed from some finite number of members of the original sequence, i.e. math s n T s n,s n 1 , dots ... of convergence, the transformed sequence is said to converge faster than the original sequence ... , assumed to be convergent. In this case, convergence acceleration is obtained. If the original sequence is divergent , the sequence transformation acts as extrapolation method to the antilimit math ... 0 k c m s n m math for some constants math c 0, dots,c k math which may depend on n , the sequence transformation math mathbf T math is called a linear sequence transformation . Sequence transformations that are not linear are called nonlinear sequence transformation s. Examples Simplest examples of linear sequence transformations include shifting all elements, math s n s n k math resp. 0 if n k 0 for a fixed k , and scalar multiplication of the sequence. A little less trivial ... sequence. The binomial transform is another linear transformation of a still more general type. An example of a nonlinear sequence transformation is Aitken s delta squared process , used to improve ... more details
In mathematics, a divisibility sequence is an integersequence math a n n in N math such that for all natural numbers m , n , math text if m mid n text then a m mid a n, math i.e., whenever one index is a multiple of another one, then the corresponding term also is a multiple of the other term. The concept can be generalized to sequences with values in any ring mathematics ring where the concept of divisibility is defined. A strong divisibility sequence is an integersequence math a n n in N math such that for all natural numbers m , n , math gcd a m,a n a gcd m,n . math Note that a strong divisibility sequence is immediately a divisibility sequence if math m mid n math , immediately math gcd m,n m math . Then by the strong divisibility property, math gcd a m,a n a m math and therefore math a m mid a n math . Examples Any constant sequence is a divisibility sequence. Every sequence of the form math a n kn math , for some nonzero integer k , is a divisibility sequence. Every sequence of the form math a n A n B n math for integers math A B 0 math is a divisibility sequence. The Fibonacci numbers F 0, 1, 1, 2, 3, 5, 8,... form a strong divisibility sequence. Elliptic divisibility sequence s are another class of such sequences. References cite book first1 Graham last1 Everest first2 Alf last2 van der Poorten first3 Igor last3 Shparlinski first4 Thomas last4 Ward title Recurrence Sequences publisher American Mathematical Society year 2003 isbn 978 0821833872 cite journal first1 Marshall last1 Hall title Divisibility sequences of third order journal Am. J. Math year 1936 pages 577&ndash 584 volume 58 jstor 2370976 cite journal first1 Morgan last1 Ward title A note on divisibility sequences journal Bull. Amer. Math. Soc volume 45 year 1939 pages 334&ndash 336 ... 243 271 External links Some http oeis.org search?q divisibility sequence divisibility sequences listed in the On line Encyclopedia of Integer Sequences . Category Sequences and series Category Integer ... more details
In mathematics , a primefree sequence is a sequence of integer s that does not contain any prime number s. More specifically, it usually means a sequence defined by the same recurrence relation as the Fibonacci number s, but with different initial conditions causing all members of the sequence to be composite number s that do not all have a common divisor . To put it algebraically, a sequence of this type ..., a sub 2 sub 1510028911088401971189590305498785 sequence OEIS A083104 A083104 in On Line Encyclopedia of Integer Sequences OEIS Graham 1964 , a sub 1 sub 62638280004239857, a sub 2 sub 49463435743205655 sequence OEIS A083105 A083105 in On Line Encyclopedia of Integer Sequences OEIS Knuth 1990 , and a sub 1 sub 407389224418, a sub 2 sub 76343678551 sequence OEIS A082411 A082411 in On Line Encyclopedia of Integer Sequences OEIS Nicol 1999 . The sequence of this type with the smallest known initial ... title Primefree Sequence urlname PrimefreeSequence Category Integer sequences Category Number theory ... > 2 there are no primes in the sequence of numbers calculated from the formula a sub ... s sequence Perhaps the best known primefree sequence is the one found by Herbert Wilf , with initial ... term of this sequence is composite relies on the periodicity of Fibonacci like number sequences modulo the members of a finite set of primes. For each prime p , the positions in the sequence ... have overlapping patterns that result in a covering set for the whole sequence. Nontriviality The requirement that the initial terms of a primefree sequence be coprime is necessary for the question to be non ... y p and more generally all subsequent value in the sequence will be multiples of p . In this case, all the numbers in the sequence will be composite, but for a trivial reason. The order of the initial ... who loved only numbers , the Wilf sequence is cited but with the initial terms switched. The resulting sequence appears primefree for the first hundred terms or so, but term 138 is the 45 digit ... more details
In general topology , a branch of mathematics, the integer broom topology , is an example of a topology on the so called integer broom space X . ref name CEIT Citation first L. A. last Steen first2 J. A. last2 Seebach title Counterexamples in Topology publisher Dover year 1995 page 140 ISBN 048668735X ref To give a set X a topology means to say which subset s of X are open set open in a manner that satisfies certain axioms ref name CEIT2 Citation first L. A. last Steen first2 J. A. last2 Seebach title Counterexamples in Topology publisher Dover year 1995 page 3 ISBN 048668735X ref The union mathematics union of open sets is an open set. The finite intersection mathematics intersection of open sets is an open set. X and the empty set are open sets. Definition of the integer broom space File Integer Broom Plot FBN.gif thumb center A subset of the integer broom center The integer broom space X is a subset of the plane R sup 2 sup . Assume that the plane is parametrised by polar coordinates . The integer broom contains the origin and the points nowrap 1 n ,&theta R sup 2 sup such that n is a non negative integer , and nowrap 1 &theta 1 k k N and k 1 . ref name CEIT The image on the right gives an illustration for nowrap 1 0 n 5 and nowrap 1 1 15 &theta 1 . Geometrically, the space consists of a series of convergent sequence s. For a fixed n , we have a sequence of points lying on circle with centre 0,0 and radius n that converges to the point n ,0 . Definition of the integer broom topology We deine a topology on X be means of a product topology . The Integer Broom space is given by the polar coordinates math n, theta in n in Z n ge 0 times theta 1 k k in Z, k ge 1 , . math Let us write nowrap 1 n ,&theta U &thinsp × &thinsp V for simplicity. The Integer Broom topology ... from R . ref name CEIT Properties The integer broom space, together with the integer broom topology ... 1995 pages 200 201 ISBN 048668735X ref References reflist DEFAULTSORT Integer Broom topology Category ... more details
values. Nevertheless for k 7 the Somos sequences contain only integer values. Several mathematicians have studied the problem of proving and explaining this integer property of the Somos sequences. ref citation first Janice L. last Malouf title An integersequence from a rational ...In mathematics , a Somos sequence is a sequence of numbers defined by a bilinear recurrence relation, discovered by mathematician Michael Somos . It is not obvious from the form of their defining recurrence that every number in a Somos sequence is an integer, but nevertheless many Somos sequences have the property that all of their members are integers. Recurrence equations A Somos k sequence, for an integer number k , is defined by the equation math a n a n k a n 1 a n k 1 a n 2 a n k 2 cdots a n k 1 2 a n k 1 2 math when k is odd, or by the analogous equation math a n a n k a n 1 a n k 1 a n 2 a n k 2 cdots a n k 2 2 math when k is even. For instance, for the first nontrivial case, k 4, the defining equation is math a n a n 4 a n 1 a n 3 a n 2 2 math while for k 5 the equation is math a n a n 5 a n 1 a n 4 a n 2 a n 3 ,. math These equations can be rearranged into the form of a recurrence relation , in which the value a sub n sub on the left hand side of the recurrence ... k are all set equal to one, as in the recurrence for the Fibonacci sequence . However it is also .... Sequence values The values in the Somos 4 sequence are 1, 1, 1, 1, 2, 3, 7, 23, 59, 314, 1529, 8209, 83313, 620297, 7869898, ... OEIS A006720 . The values in the Somos 5 sequence are 1, 1, 1, 1, 1, 2, 3, 5, 11, 37, 83, 274, 1217, 6161, 22833, 165713, ... OEIS A006721 . The values in the Somos 6 sequence ... in the Somos 7 sequence are 1, 1, 1, 1, 1, 1, 1, 3, 5, 9, 17, 41, 137, 769, 1925, 7203, 34081 ... Jim Propp s Somos Sequence Site mathworld title Somos Sequence urlname SomosSequence Category Integer sequences ... more details
Unreferenced date December 2009 In mathematics , an ergodic sequence is a certain type of integersequence , having certain equidistribution properties. Definition Let math A a j math be an infinite, strictly increasing sequence of positive integers. Then, given an integer q , this sequence is said to be ergodic mod q if, for all integers math 1 leq k leq q math , one has math lim t to infty frac N A,t,k,q N A,t frac 1 q math where math N A,t mbox card a j in A a j leq t math and cardinality card is the count the number of elements of a set, so that math N A,t math is the number of elements in the sequence A that are less than or equal to t , and math N A,t,k,q mbox card a j in A a j leq t, , a j mod q k math so math N A,t,k,q math is the number of elements in the sequence A , less than t , that are equivalent to k modulo q . That is, a sequence is an ergodic sequence if it becomes uniformly distributed mod q as the sequence is taken to infinity. An equivalent definition is that the sum math lim t to infty frac 1 N A,t sum j a j leq t exp frac 2 pi ika j q 0 math vanish for every integer k with math k mod q ne 0 math . If a sequence is ergodic for all q , then it is sometimes said to be ergodic for periodic systems . Examples The sequence of positive integers is ergodic for all q . Almost all Bernoulli sequence s, that is, sequences associated with a Bernoulli process , are ergodic for all q . That is, let math Omega,Pr math be a probability space of random variable s over two letters math 0,1 math . Then, given math omega in Omega math , the random variable math X j omega math is 1 with some probability p and is zero with some probability 1 p this is the definition of a Bernoulli process. Associated with each math omega math is the sequence of integers math mathbb Z omega n in mathbb Z X n omega 1 math Then almost every sequence math mathbb Z omega math is ergodic. See ... Sequence Category Ergodic theory Category Sequences and series ... more details
In mathematics, the Golomb sequence , named after Solomon W. Golomb but also called Silverman s sequence , is a non decreasing integersequence where a sub n sub is the number of times that n occurs in the sequence, starting with a sub 1 sub 1, and with the property that for n 1 each a sub n sub is the unique integer which makes it possible to satisfy the condition. For example, a sub 1 sub 1 says that 1 only occurs once in the sequence, so a sub 2 sub cannot be 1 too, but it can be, and therefore must be, 2. The first few values are 1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12 OEIS id A001462 . Colin Mallows has given an explicit recurrence relation a 1 1 a n 1 1 a n 1 &minus a a n . An asymptotic expression for a sub n sub is math varphi 2 varphi n varphi 1 , math where is the golden ratio . References Richard K. Guy , Unsolved Problems in Number Theory 3rd ed , Springer Verlag , 2004 ISBN 0 387 20860 7 section E25. Numtheory stub Category Integer sequences Category Golden ratio fr Suite de Golomb nl Rij van Golomb ... more details
are given. See also Thue Morse Sequence External links http oeis.org On Line Encyclopedia of Integer ...In mathematics , a fractal sequence is one that contains itself as a proper subsequence. An example is 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, ... If the first occurrence of each n is deleted, the remaining sequence is identical to the original. The process can be repeated indefinitely, so that actually, the original sequence contains not only one copy of itself, but rather, infinitely many. Definition The precise definition of fractal sequence depends on a preliminary definition a sequence x x sub n sub is an infinitive sequence if for every i, F1 x sub n sub i for infinitely many n. Let a i,j be the jth index n for which x sub n sub i. An infinitive sequence x is a fractal sequence if two additional conditions hold F2 if i 1 x sub n sub , then there exists m n such that i x sub m sub F3 if h i then for every j there is exactly one k such that a i,j a h,k a i,j 1 . According to F2 , the first occurrence of each i 1 in x must be preceded at least once by each of the numbers 1, 2, ..., i 1, and according to F3 , between consecutive occurrences of i in x, each h less than i occurs exactly once. Example Suppose is a positive irrational number. Let S the set of numbers c d , where c and d are positive integers and let c sub n sub d sub n sub be the sequence obtained by arranging the numbers in S in increasing order. The sequence c sub n sub is the signature of , and it is a fractal sequence. For example, the signature of the golden ratio i.e., 1 sqrt 5 2 begins with 1, 2, 1, 3, 2, 4, 1, 3, 5, 2, 4, 1, 6, 3, 5, 2, 7, 4, 1, 6, 3, 8, 5, ... and the signature of 1 1 begins with 1, 1, 2, 1, 2, 1, 3, 2, 1, 3, 2, 4, 1, 3, 2, 4, 1, 3, 2, 4, 1, 3, 5, ... These are sequences OEIS2C A084532 and OEIS2C A084532 in the On Line Encyclopedia of Integer ... Sequences and Interspersions, Ars Combinatoria 45 1997 157 168. Category Fractals Category Integer ... more details
, A Recursive Formula for the Kolakoski Sequence A000002, Journal of Integer Sequences 9 2006 Article ...Inappropriate tone date August 2008 In mathematics , the Kolakoski sequence named after William Kolakoski ... this sequence are as follows 1 The only numbers allowed are 1 and 2 2 Each number tells us how many numbers to append to the sequence 2a 1 tells us to append one more number 2b 2 tells us to append two more numbers 3 We can have no more than two of the same number in sequence 3a This is because if we write 222 that means that somewhere in the sequence it told us to append 3 twos, but the only numbers ... A sub 0 sub 1 This is an example of a self generating sequence, with the first term as 1. Each individual ... the sequence we have to write one 2 after the 1 because the 1 tells us to write one number, and because ... sequence so far 1 2 . Next, The 2 tells us that the length, or the run, of the next set of numbers must be 2, but we cannot have any more than two ones or two twos in sequence because that would mean that the sequence told us to write three 2 s, which would mean a 3 would have to be in the sequence, which is not allowed. Therefore, we must write 2 1. This gives us our sequence so far 1 2 2 1 . The next ... write is a 1 however, we cannot write three 1 s in a row, so we must append 1 2 to the sequence. This gives us our sequence so far 1 2 2 1 1 2 . The fourth number in the list is a 1. As we wrote a 2 last we write a 1 now. This gives us our sequence so far 1 2 2 1 1 2 1 . The fifth number in the sequence is a 1. As we wrote a 1 before, we must write a 2 now. This gives us our sequence so far 1 2 2 1 1 2 1 2 . This continues on. Another explanation for the generation of the Kolakoski sequence is indicated ... 5 so far... 1,2,2,1,1,2,1 continue generating forever. To truly understand how the sequence is generated ... unsolved problems are presented at http faculty.evansville.edu ck6 integer index.html Integer Sequences ... 1993 93 84.html Notes on the Kolakoski Sequence , Chv tal proved that the upper density of 1 ... more details
An automatic sequence or k automatic sequence is an infinite sequence of terms characterized by a finite automaton . The n th term of the sequence is a mapping of the final state of the automaton when its input is the digits of n in some fixed base k . ref name as1 Allouche & Shallit 2003 p.152 ref A k automatic set is a set of non negative integers for which the sequence of values of its characteristic function is an automatic sequence that is, membership of n in the set can be determined by a finite state automaton on the digits of n in base k . ref Allouche & Shallit 2003 p.168 ref Automaton point of view Let q be an integer , and A E , , e be a deterministic automaton where E is the finite Set mathematics set of State computer science state s E 0, q 1 E is the transition function math e in E math is the initial state also let A be a finite set, and E A a Projection mathematics projection towards A . For each n , take m n e , math n math where math n math is n written in base q . Then the sequence m m 1 m 2 m 3 ... is called a q automatic sequence . ref name as1 Substitution point of view Let be a morphism of the free monoid E sup sup with math sigma E subseteq ... is a q automatic sequence over A . ref Allouche & Shallit 2003 p.175 ref 1 automatic sequences k automatic ... by defining a 1 automatic sequence to be a sequence whose n th term depends on the unary numeral system ... Examples The following sequences are automatic Thue Morse sequence take E A 0, 1 , e 0, id, and ... base 2 representation of n and the sequence is thus 2 automatic. ref name as1 Rudin Shapiro sequence Baum Sweet sequence Regular paperfolding sequence Automatic real number An automatic real number is a real number for which the base b expansion is an automatic sequence. ref name hejhal556 ... in integer bases , Ann. of Math. 165 2 2007 , pp. 547 565. ref References reflist Jean Paul ... Press 1988, ISBN 0521335450 DEFAULTSORT Automatic Sequence Category Combinatorics on words ... more details
In mathematics , an integersequence is called a complete sequence if every positive integer can be expressed as a sum of values in the sequence, using each value at most once. For example, the sequence of powers of two 1, 2, 4, 8, ... , based on the binary numeral system , is a complete sequence given ... to encode an integer from the complete sequence and a minimizing algorithm that will attempt to maximize the number of terms needed to encode the same integer. A coding for the sequence of the number ... and sum them to obtain that number e.g. 37 100101 sub 2 sub 1 4 32 . This sequence is minimal, since ... produces only even numbers, no odd number can be formed. Powers of three no integer having a digit ... of generality, assume the sequence a sub n sub is in nondecreasing order, and define the partial sums ... a k 1 , forall , k ge 1 math are both necessary and sufficient for a sub n sub to be a complete sequence ... , k ge 1 math are sufficient for a sub n sub to be a complete sequence. ref name Honsberger However ... sequences Below is a list of the better known complete sequences. The sequence of the number 1 ... above, n 3 gives the Generalizations of Fibonacci numbers Tribonacci numbers etc. The Lazy caterer s sequence ... lines as dividers. All higher dimensions of the Lazy caterer s sequence that uses n hyperplanes of dimension d 1 to maximally divide a space of dimension d . OEIS id A145111 The Cookie cutter s sequence ... dimensions of the Cookie cutter s sequence that uses n hyperspherical surfaces of dimension d 1 ... form a complete sequence due to the binary numeral system, in fact any complete sequence can be used ... of the sequence the next rightmost to the next member and so on. Bits set to 1 are included in the sum ... system, based on the Fibonacci sequence, the number 17 can be encoded in six different ways ... systems Other coding systems can be similarly devised. As with the Fibonacci sequence above, these coding ... for the sequence of the number 1 followed by the prime numbers using a minimizing algorithm ... more details
In mathematics , a Hofstadter sequence is a member of a family of related integer sequences defined by non ... of Hofstadter s Q Sequence journal Journal of Integer Sequences volume 10 issue 7 publisher University ... and background Figure Figure sequence and chapter V on recursive structures and processes remaining ... FigureSequence title Hofstadter Figure Figure Sequence ref math begin align R 1 & 1 S 1 2 R n & R n 1 S n 1 , quad n 1. end align math with the sequence S n defined as the positive integers ..., 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, ... OEIS id A030124 Hofstadter G sequence The Hofstadter G sequence is defined as follows ref name hof137 Hofstadter 1980 p137 ref ref mathworld urlname HofstadterG Sequence title Hofstadter G Sequence ref math begin align G 0 & 0 G n & n G G n 1 , quad n 0. end align math The first few terms of this sequence are 0, 1, 1, 2, 3, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 9, 10, 11, 11, 12, 12, ... OEIS id A005206 Hofstadter H sequence The Hofstadter H sequence is defined as follows ref name hof137 ref mathworld urlname HofstadterH Sequence title Hofstadter H Sequence ... of this sequence are 0, 1, 1, 2, 3, 4, 4, 5, 5, 6, 7, 7, 8, 9, 10, 10, 11, 12, 13, 13, 14, ... OEIS ..., 7, 8, 9, 9, 10, 11, 11, 12, 12, ... OEIS id A005379 Hofstadter Q sequence The Hofstadter Q sequence is defined as follows ref name hof137 ref mathworld urlname HofstadtersQ Sequence title Hofstadter s Q Sequence ref math begin align Q 1 & Q 2 1, Q n & Q n Q n 1 Q n Q n 2 , quad n 2. end align math The first few terms of the sequence are 1, 1, 2, 3, 3, 4, 5, 5, 6, 6, 6, 8, 8, 8, 10, 9, 10, 11, 11, 12, ... OEIS id A005185 Hofstadter named the terms of the sequence Q numbers ref name hof137 thus the Q number of 6 is 4. The presentation of the Q sequence in Hofstadter s book is actually the first known mention of a meta Fibonacci sequence in literature. ref Emerson 2006 p1, p7 ref While the terms of the Fibonacci sequence are determined by summing the two preceding terms, the two preceding terms ... more details
Orphan date December 2009 A disjunctive sequence is an infinite Sequence Infinite sequences in theoretical computer science sequence over a finite alphabet computer science alphabet of character computing characters in which every String computer science Formal theory finite string appears as a substring . For instance, the binary Champernowne constant Champernowne sequence math 0 1 00 01 10 11 000 001 ldots math formed by concatenating all binary strings in shortlex order , clearly contains all the binary strings and so is disjunctive. The spaces above are not significant and are present solely to make clear the boundaries between strings . Any normal number normal sequence a sequence in which each string of equal length appears with equal frequency is disjunctive, but the Conversion logic converse is not true. For example, letting 0 sup n sup denote the string of length n consisting of all 0s, consider the sequence math 0 0 1 1 0 2 00 0 4 01 0 8 10 0 16 11 0 32 000 0 64 ldots math obtained by splicing exponentially long strings of 0s into the shortlex order ing of all binary strings. Most of this sequence consists of long runs of 0s, and so it is not normal, but it is still disjunctive. Examples The following result ref citation last1 Calude first1 C. author1 link Cristian S. Calude last2 Priese first2 L. author2 link Lutz Priese last3 Staiger first3 L. author3 link Ludwig Staiger publisher University of Auckland, New Zealand pages 1 35 title Disjunctive sequences An overview ... sequence of positive integers such that Limit of a sequence tt lim tt sub n sub a sub n 1 sub a sub n sub 1, then for any positive integer m and any integer Radix base b 2, there is an a sub n ... sequence obtained by concatenating the base b expressions for a sub 1 sub , a sub 2 sub , a sub 3 ... a sub n sub n sup k sup , where k is a fixed positive integer. In this case, tt lim tt sub n sub a sub ... Sequence Category Sequences and series ... more details
In Intuitionism intuitionistic mathematics , a choice sequence is a Constructivism mathematics constructive formulation of a sequence . Since the Intuitionistic school of mathematics, as formulated by L. E. J. Brouwer , rejects the idea of a completed infinity , in order to use a sequence which is, in classical ... object which can serve the same purpose as a sequence. Thus, Brouwer formulated the choice sequence ... A distinction is made between lawless and lawlike sequences. A lawlike sequence is one that can ..., the natural numbers math mathbb N math can be thought of as a lawlike sequence the sequence ... successor function . Given this formulation, we know that the math i math th element in the sequence ... effectively determines the value for any argument it takes, and thus describes a lawlike sequence. A lawless also, free sequence, on the other hand, is one that is not predetermined. It is to be thought of as a procedure for generating values for the arguments 0, 1, 2, .... That is, a lawless sequence ... of the sequence math alpha math such that At any given moment of construction of the sequence math alpha math , only an initial segment of the sequence is known, and no restrictions are placed ... is slightly misleading, as we may specify, for example, that the values in a sequence be drawn exclusively ... of the sequence. The canonical example of a lawless sequence is the series of rolls of a dice die ... rolls for math k in mathbb N math . Further, we restrict the values of the sequence to be in the set math 1,2,3,4,5,6 math . This specification comprises the procedure for generating the lawless sequence in question. At no point, then, is any particular future value of the sequence known. Axiomatization ... above. Let math alpha in n math denote the relation the sequence math alpha math begins with the initial sequence math n math for choice sequence math alpha math and finite segment math n math more specifically, math n math will probably be an integer Numbering computability theory encoding a finite ... more details
multiple image align right direction vertical width 250 image1 Cauchy sequence illustration.svg caption1 a The plot of a Cauchy sequence math x n , math shown in blue, as math x n math versus math n math If the space containing the sequence is complete, the ultimate destination of this sequence that is, the limit exists. image2 Cauchy sequence illustration2.svg caption2 b A sequence that is not Cauchy. The elements of the sequence fail to get arbitrarily close to each other as the sequence progresses. In mathematics , a Cauchy sequence pronounced IPA ko i , named after Augustin Louis Cauchy , is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite number of elements of the sequence are less ... metric space one where all such sequences are known to limit of a sequence converge to a limit , the criterion for convergence depends only on the terms of the sequence itself. This is often ... relatively easily to produce a Cauchy sequence, consisting of the iterates. The notions above are not as unfamiliar ... number x has a decimal expansion is an implicit acknowledgment that a particular Cauchy sequence of rational ... exist in the form of Cauchy filter and Cauchy net . In Real numbers A sequence math x 1, x 2, x 3, ldots ... number , there is a positive integer N such that for all natural numbers m , n N math x m x n varepsilon ... x m math and math x n math . Formally, given a metric space X , d , a sequence math x 1, x 2, x 3, ldots math is Cauchy, if for every positive real number 0 there is a positive integer N such that for all ... of the sequence are getting closer and closer together in a way that suggests that the sequence ought to have a limit of a sequence limit in X . Nonetheless, such a limit does not always exist within X . Completeness A metric space X in which every Cauchy sequence converges to an element of X ... . Any Cauchy sequence of elements of X must be constant beyond some fixed point, and converges to the eventually ... more details
In recreational mathematics a juggler sequence is an integersequence that starts with a positive integer a sub 0 sub , with each subsequent term in the sequence defined by the recurrence relation math a k 1 begin cases left lfloor a k frac 1 2 right rfloor, & mbox if a k mbox is even left lfloor a k frac 3 2 right rfloor, & mbox if a k mbox is odd end cases math Background Juggler sequences were publicised by American mathematician and author Clifford A. Pickover . ref cite book last Pickover first Clifford A. authorlink Clifford A. Pickover title Computers and the Imagination publisher St. Martin s Press date 1992 pages Chapter 40 isbn 978 0312083434 ref The name is derived from the rising and falling nature of the sequences, like balls in the hands of a juggler . ref cite book last Pickover first Clifford A. authorlink Clifford A. Pickover title The Mathematics of Oz publisher Cambridge University Press date 2002 pages Chapter 45 isbn 978 0521016780 ref For example, the juggler sequence starting with a sub 0 sub 3 is math a 1 lfloor 3 frac 3 2 rfloor lfloor 5.196 dots rfloor 5, math math a 2 lfloor 5 frac 3 2 rfloor lfloor 11.180 dots rfloor 11, math math a 3 lfloor 11 frac 3 2 rfloor ... lfloor 1.414 dots rfloor 1. math If a juggler sequence reaches 1, then all subsequent terms are equal ... for initial terms up to 10 sup 6 sup , ref MathWorld title Juggler Sequence urlname JugglerSequence ... sequence starting at n takes to first reach 1, and h n to be the maximum value in the juggler sequence starting at n . For small values of n we have class wikitable n Juggler sequence l n OEIS id A007320 ... before descending to 1. For example, the juggler sequence starting at a sub 0 sub 37 reaches a maximum value of 24906114455136. Harry J. Smith has determined that the juggler sequence starting at a sub ... External links http members.chello.nl k.ijntema juggler.html Juggler sequence calculator at Collatz ... index.html Juggler Number pages by Harry J. Smith Category Integer sequences Category Unsolved ... more details
In mathematics , an aliquot sequence is a recursive sequence in which each term is the sum of the proper divisor s of the previous term. The aliquot sequence starting with a positive integer k can be defined formally in terms of the divisor function sum of divisors function sub 1 sub in the following way ref name mw MathWorld urlname AliquotSequence title Aliquot Sequence ref s sub 0 sub k s sub n sub sub 1 sub s sub n &minus 1 sub &minus s sub n &minus 1 sub . For example, the aliquot sequence of 10 is 10, 8, 7, 1, 0 because sub 1 sub 10 &minus 10 5 2 1 8 sub 1 sub 8 &minus 8 4 2 1 7 ... an aliquot sequence might not terminate A perfect number has a repeating aliquot sequence of period 1. The aliquot sequence of 6, for example, is 6, 6, 6, 6, ... An amicable number has a repeating aliquot sequence of period 2. For instance, the aliquot sequence of 220 is 220, 284, 220, 284, ... A sociable number has a repeating aliquot sequence of period 3 or greater. Sometimes the term sociable number is used to encompass amicable numbers as well. For instance, the aliquot sequence of 1264460 is 1264460, 1547860, 1727636, 1305184, 1264460, ... Some numbers have an aliquot sequence which ..., the aliquot sequence of 95 is 95, 25, 6, 6, 6, 6, ... . Numbers like 95 that are not perfect, but have an eventually repeating aliquot sequence of period 1 are called aspiring numbers OEIS2C id A063769 ... is that every aliquot sequence ends in one of the above ways with a prime number, a perfect number ... Catalan s Aliquot Sequence Conjecture ref The alternative would be that a number exists whose aliquot sequence is infinite, yet aperiodic . There are several numbers whose aliquot sequences As of 2010 ... Others Aliquot000.htm Aliquot sequence summary page for sequences up to 100000 there are similar ... Varona Paul Zimmermann. http www.expmath.org expmath volumes 11 11.2 3630finishes1.pdf Aliquot Sequence ... navbox DEFAULTSORT Aliquot Sequence Category Arithmetic functions Category Divisor function da Alikvotf lge ... more details
In mathematics , a bounded sequence s sub 1 sub , s sub 2 sub , s sub 3 sub , of real number s is said ... theory and have applications to Monte Carlo integration . Definition A bounded sequence s sub ... n sub c , d denotes the number of elements, out of the first n elements of the sequence, that are between c and d . For example, if a sequence is equidistributed in 0, 2 , since the interval ... of the first n members of the sequence which fall between 0.5 and 0.9 must approach 1 5. Loosely speaking, one could say that each member of the sequence is equally likely to fall anywhere in its range. However, this is not to say that s sub n sub is a sequence of random variables rather, it is a determinate sequence of real numbers. Discrepancy We define the discrepancy D N for a sequence ... A sequence is thus equidistributed if the discrepancy D N tends to zero as N tends to infinity. Equidistribution is a rather weak criterion to express the fact that a sequence fills the segment leaving ... in the segment, but there will be large gaps compared to a sequence which first enumerates multiples ... this for smaller and smaller values of . See low discrepancy sequence for stronger criteria and constructions .... Equidistribution modulo 1 The sequence a sub 1 sub , a sub 2 sub , a sub 3 sub , is said to be equidistributed modulo 1 or uniformly distributed modulo 1 if the sequence of the fractional parts ... 1 . Examples The sequence of all multiples of an irrational , 0, &alpha , 2&alpha , 3&alpha , 4 ..., if p is a polynomial with at least one irrational coefficient other than the constant term then the sequence ... of Johannes van der Corput . The sequence log n is not uniformly distributed modulo 1. The sequence ... number theory , proved by I. M. Vinogradov in 1935. The van der Corput sequence is equidistributed ... int 0 1 f x , dx. math ol start 3 li For every nonzero integer k , math lim n to infty frac 1 n sum ... the behaviour of a parametrised sequence for almost all values of some parameter that is, for values ... more details
infobox website name On Line Encyclopedia of Integer Sequences logo File OEISicon.svg 150px url http oeis.org oeis.org launch date start date 1996 author Neil Sloane The On Line Encyclopedia of Integer Sequences OEIS , also cited simply as Sloane s , is an online database of integersequence s, created ... of Integer Sequences ref In 2004, Sloane celebrated the addition of the 100,000th sequence to the database ... on integer sequences of interest to both professional mathematician s and recreational mathematics ... database of its kind. Each entry contains the leading terms of the sequence, Keyword computer ... to generate a Graph of a function graph or play a Computer music musical representation of the sequence ... Sloane started collecting integer sequences as a graduate student in 1965 to support his work in combinatorics ... in book form twice A Handbook of Integer Sequences 1973, ISBN 0 12 648550 X , containing 2,372 sequences in lexicographic order and assigned numbers from 1 to 2372. The Encyclopedia of Integer ... which may differ in their few initial terms in A Handbook of Integer Sequences as N numbers from N0001 ... work, Sloane founded the Journal of Integer Sequences in 1998. ref http www.cs.uwaterloo.ca journals JIS Journal of Integer Sequences ISSN 1530 7638 ref The database continues to grow at a rate ... 2010 11 17 author Neil Sloane ref The 200,000th sequence, OEIS2C id A200000 , was added to the database ... Charles Greathouse to choose a special sequence for A200000. ref cite web url http oeis.org wiki Suggested ... integer sequences, the OEIS also catalogs sequences of fraction mathematics fraction s, the digits of transcendental number s, complex number s and so on by transforming them into integer sequences. Sequences of rationals are represented by two sequences named with the keyword frac the sequence of numerators and the sequence of denominators. For example, the fifth order Farey sequence , math textstyle ... 5 math , is catalogued as the numerator sequence 1, 1, 1, 2, 1, 3, 2, 3, 4 OEIS2C id A006842 and the denominator ... more details
of the function f x x sup 5 sup cx sup 4 sup is a Sidon sequence, where . denotes integer ...In number theory , a Sidon sequence or Sidon set , named after the Hungarian mathematician Simon Sidon , is a sequence A a sub 0 sub , a sub 1 sub , a sub 2 sub , ... of natural numbers in which all pairwise sums a sub i sub a sub j sub i j are different. Sidon introduced the concept in his investigations of Fourier series . The main problem, posed by Sidon, ref citation first1 P. last1 Erd s author1 link Paul Erd s first2 P. last2 Tur n author2 link P l Tur n title On a problem of Sidon in additive number theory and on some related problems journal J. London Math. Soc. volume 16 year 1941 pages 212 215 doi 10.1112 jlms s1 16.4.212 url http www.renyi.hu p erdos 1941 01.pdf . http www.math inst.hu p erdos 1944 02.pdf Addendum , 19 1944 , 208. ref is how many elements can A have up to some number x . Despite a large body of research ref citation ... bound. If, however, we consider an infinite Sidon sequence A and let A x denote the number of its ... sequence with math A x c sqrt 3 x math for every x . Mikl s Ajtai Ajtai , J nos Koml s mathematician ... E. last3 Szemer di author3 link Endre Szemer di title A dense infinite Sidon sequence journal European Journal of Combinatorics volume 2 year 1981 pages 1 11 mr 0611925 issue 1 . ref of a Sidon sequence ... , who proved ref citation first I. Z. last Ruzsa authorlink Imre Z. Ruzsa title An infinite Sidon sequence ... . ref that a Sidon sequence with math A x x sqrt 2 1 o 1 math exists. Erd s conjectured that an infinite ... of such a sequence a sub 0 sub , a sub 1 sub ,...with the weaker property that for every ... c . Erd s further conjectured that there exists a nonconstant integer coefficient polynomial whose values at the natural numbers form a Sidon sequence. Specifically, he asked if the set of fifth ... Sidon Sequence Category Number theory Category Combinatorics hu Sidon sorozat vi D y Sidon ... more details
An M sequence may refer to Regular sequence algebra Regular sequence , which is an important topic in commutative algebra. A maximum length sequence , which is a type of pseudorandom binary sequence. disambig ... more details
A Ducci sequence is a sequence of n tuple n tuples of integer s. Given an n tuple of integers math a 1,a 2,...,a n math , the next n tuple in the sequence is formed by taking the absolute differences of neighbouring integers math a 1,a 2,...,a n rightarrow a 1 a 2 , a 2 a 3 , ..., a n a 1 , . math Another way of describing this is as follows. Arrange n integers in a circle and make a new circle by taking the difference between neighbours, ignoring any minus signs then repeat the operation. Ducci sequences are named after Enrico Ducci, the Italian mathematician credited with their discovery. Ducci sequences are also known as the Ducci map or the n number game . Open problems in the study of these maps still remain. ref name Chamberland&Thomas2004 cite journal last1 Chamberland first1 Marc last2 ... the second n tuple onwards, it is clear that every integer in each n tuple in a Ducci sequence is greater ... 2002 breuer homepage research.html Ducci Sequence http www.cut the knot.org SimpleGames IntIter.shtml Integer Iterations on a Circle at Cut the Knot DEFAULTSORT Ducci Sequence Category Sequences ..., the sequence of n tuples must sooner or later repeat itself. Every Ducci sequence therefore eventually becomes Periodic function periodic . If n is a power of 2 every Ducci sequence eventually ..., a Ducci sequence will either eventually reach an n tuple of zeros or will settle into a periodic ..., a Ducci sequence starting with the n tuple 1, q , q sup 2 sup , q sup 3 sup where q is the irrational ... loop. The 4 tuple sequence starting with 0, 653, 1854, 4063 takes 24 iterations to reach ..., 128, 128, 128 rightarrow 0, 0, 0, 0 math This 5 tuple sequence enters a period 15 binary loop after ... 6 tuple sequence shows that sequences of tuples whose length is not a power of two may still reach ... the sequence in modulo two. That is ref name Breauer Florian Breuer, Ducci sequences in higher dimensions ... the basis for proving when the sequence vanish to all zeros. Cellular automata File CA rule ... more details
The Padovan sequence is the sequence of integer s P n defined by the initial values math P 0 P 1 P 2 ... the Padovan sequence. The Padovan sequence is named after Richard Padovan who attributed its discovery ... Primitive . The sequence was described by Ian Stewart mathematician Ian Stewart in his Scientific American ... and by MathWorld . Other sources may start the sequence at a different place, in which case some ... sequence also satisfies the recurrence relation math P n P n 1 P n 5 math Starting from this, the defining ... Perrin sequence satisfies the same recurrence relations as the Padovan sequence, although it has different initial values. This is a property of recurrence relations. The Perrin sequence can be obtained from the Padovan sequence by the following formula math mathrm Perrin n P n 1 P n 10 . , math Extension to negative parameters As with any sequence defined by a recurrence relation, Padovan ... sequence is 2 less than P n 5 i.e. math sum m 0 n P m P n 5 2. math Sums of alternate terms, sums of every third term and sums of every fifth term are also related to other terms in the sequence ... in the Padovan sequence satisfy the following identities math sum m 0 n P m 2 P n 2 2 P n 1 2 P n 3 ... Other identities The Padovan sequence also satisfies the identity math P n 2 P n 1 P n 1 P n 7 . , math The Padovan sequence is related to sums of binomial coefficient s by the following identity math ... sequence numbers can be written in terms of powers of the roots of the equation math x 3 x 1 0. , math ... roots q and r . Given these three roots, the Padovan sequence can be expressed by a formula involving ... is the integer closest to math frac p n 1 s math , where s p a 1.0453567932525329623... is the only ... 0. The ratio of successive terms in the Padovan sequence approaches p , which has a value of approximately 1.324718. This constant bears the same relationship to the Padovan sequence and the Perrin sequence as the golden ratio does to the Fibonacci sequence. Combinatorial interpretations P n ... more details
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