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The Encyclopedia of Integer Sequences
"... This article gives a brief introduction to the OnLine Encyclopedia of Integer Sequences (or OEIS). The OEIS is a database of nearly 90,000 sequences of integers, arranged lexicographically. The entry for a sequence lists the initial terms (50 to 100, if available), a description, formulae, programs ..."
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This article gives a brief introduction to the OnLine Encyclopedia of Integer Sequences (or OEIS). The OEIS is a database of nearly 90,000 sequences of integers, arranged lexicographically. The entry for a sequence lists the initial terms (50 to 100, if available), a description, formulae, programs to generate the sequence, references, links to relevant web pages, and other
Permutations of the Natural Numbers with Prescribed Difference Multisets
 Integer: Electronic Journal of Combinatorial Number Theory
, 2006
"... We study permutations π of the natural numbers for which the numbers π(n) are chosen greedily under the restriction that the differences π(n)−n belong to a given (multi)subset M of Z for all n ∈ S, a given subset of N. Various combinatorial properties of such permutations (for quite general M and S) ..."
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We study permutations π of the natural numbers for which the numbers π(n) are chosen greedily under the restriction that the differences π(n)−n belong to a given (multi)subset M of Z for all n ∈ S, a given subset of N. Various combinatorial properties of such permutations (for quite general M and S) are exhibited and others conjectured. Our results generalise to a large extent known facts in the case M = Z, S = N, where the permutation π arises in the study of the game of Wythoff Nim. 1.
Complementary equations and Wythoff sequences
 J. Integer Seq
"... The lower Wythoff sequence a = (a(n)) and upper Wythoff sequence b = (b(n)) are solutions of many complementary equations f(a, b) = 0. Typically, f(a, b) involves composites such as a(a(n)) and a(b(n)), and each such sequence is treated as a binary word (e.g., aa and ab). Conversely, each word repr ..."
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The lower Wythoff sequence a = (a(n)) and upper Wythoff sequence b = (b(n)) are solutions of many complementary equations f(a, b) = 0. Typically, f(a, b) involves composites such as a(a(n)) and a(b(n)), and each such sequence is treated as a binary word (e.g., aa and ab). Conversely, each word represents a sequence and, as such, is a linear combination of a, b, and 1, in which the coefficients of a and b are consecutive Fibonacci numbers. For example, baba = 3a + 5b − 6. 1
RESTRICTIONS OF mWYTHOFF NIM AND pCOMPLEMENTARY BEATTY SEQUENCES
, 2009
"... Fix a positive integer m. The game of mWythoff Nim (A.S. Fraenkel, 1982) is a wellknown extension of Wythoff Nim (W.A. Wythoff, 1907). The set of Ppositions may be represented as a pair of increasing sequences of nonnegative integers. It is wellknown that these sequences are socalled complem ..."
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Cited by 6 (4 self)
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Fix a positive integer m. The game of mWythoff Nim (A.S. Fraenkel, 1982) is a wellknown extension of Wythoff Nim (W.A. Wythoff, 1907). The set of Ppositions may be represented as a pair of increasing sequences of nonnegative integers. It is wellknown that these sequences are socalled complementary Beatty sequences, that is they satisfy Beatty’s theorem. For a positive integer p, we generalize the solution of mWythoff Nim to a pair of pcomplementary—each nonnegative integer is represented exactly p times—Beatty sequences a = (an)n∈N0 and b = (bn)n∈N0, which, for all n, satisfy bn − an = mn. Our main result is that {{an, bn}  n ∈ N0} represents the solution to three new ’prestrictions’ of mWythoff Nim—of which one has a certain blocking manoeuvre on the rooktype options. C. Kimberling has shown that the solution of Wythoff Nim satisfies the complementary equation xxn = yn − 1. We generalize this formula to a certain ’pcomplementary equation’ satisfied by our pair a and b. Further, if p> 1, we prove that this pair is unique in the sense that it is the only pair of pcomplementary Beatty sequences of which one of the sequences is strictly increasing. We also show that one may obtain our new pair of sequences by three socalled Minimal EXclusive algorithms.
The first column of an interspersion
 Fib. Quart
, 1994
"... In 1977, K B. Stolarsky [9] introduced an array of positive integers whose first row consists of the Fibonacci numbers {Fn:«>2}:1 2 3 5 8 13.... The subsequent rows are "generalized Fibonacci sequences. " In fact, much more is true. The rows of the array are, in a sense, the set of all ..."
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In 1977, K B. Stolarsky [9] introduced an array of positive integers whose first row consists of the Fibonacci numbers {Fn:«>2}:1 2 3 5 8 13.... The subsequent rows are "generalized Fibonacci sequences. " In fact, much more is true. The rows of the array are, in a sense, the set of all "positive Fibonacci sequences " of integers. This fact was proved by D. Morrison [7], who
Journal of Integer Sequences, Vol. 10 (2007), Article 07.5.1 Interspersions and Fractal Sequences Associated with Fractions c j /d k
"... Suppose c ≥ 2 and d ≥ 2 are integers, and let S be the set of integers ⌊ c j /d k ⌋ , where j and k range over the nonnegative integers. Assume that c and d are multiplicatively independent; that is, if p and q are integers for which c p = d q, then p = q = 0. The numbers in S form interspersions in ..."
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Suppose c ≥ 2 and d ≥ 2 are integers, and let S be the set of integers ⌊ c j /d k ⌋ , where j and k range over the nonnegative integers. Assume that c and d are multiplicatively independent; that is, if p and q are integers for which c p = d q, then p = q = 0. The numbers in S form interspersions in various ways. Related fractal sequences and permutations of the set of nonnegative integers are also discussed. 1
Fractal Sequences and Restricted Nim
"... The Grundy number of an impartial game G is the size of the unique Nim heap equal to G. We introduce a new variant of Nim, Restricted Nim, which restricts the number of stones a player may remove from a heap in terms of the size of the heap. Certain classes of Restricted Nim are found to produce seq ..."
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The Grundy number of an impartial game G is the size of the unique Nim heap equal to G. We introduce a new variant of Nim, Restricted Nim, which restricts the number of stones a player may remove from a heap in terms of the size of the heap. Certain classes of Restricted Nim are found to produce sequences of Grundy numbers with a selfsimilar fractal structure. Extending work of C. Kimberling, we obtain new characterizations of these “fractal sequences ” and give a bijection between these sequences and certain uppertriangular arrays. As a special case we obtain the game of Serial Nim, in which the Nim heaps are ordered from left to right, and players can move only in the leftmost nonempty heap. 1
Some Very Interesting Sequences
, 2006
"... First of all, we hope that anyone reading this is a Nerd. (Certainly the authors are, and quite proud of it!) Nerds have been interested in sequences of numbers ..."
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First of all, we hope that anyone reading this is a Nerd. (Certainly the authors are, and quite proud of it!) Nerds have been interested in sequences of numbers