A035615 Number of winning length n strings with a 2-symbol alphabet in "same game".

1, 0, 2, 2, 6, 12, 26, 58, 126, 278, 602, 1300, 2774, 5878, 12350, 25778, 53470, 110332, 226610, 463602, 945214, 1921550, 3896642, 7885092, 15927086, 32121582, 64697726, 130166378, 261637446, 525478668, 1054673162, 2115601450, 4241716734, 8501080838, 17031744170

Offset

0,3

Comments

Strings that can be reduced to null string by repeatedly removing an entire run of two or more consecutive symbols.

Links

Robert Price, Table of n, a(n) for n = 0..1000

Chris Burns and Benjamin Purcell, A note on Stephan's conjecture 77, preprint, 2005. [Cached copy]

Chris Burns and Benjamin Purcell, Counting the number of winning strings in the 1-dimensional same game, Fibonacci Quarterly, 45(3) (2007), 233-238.

Sascha Kurz, Polynomials for same game, pdf.

Ralf Stephan, Prove or disprove: 100 conjectures from the OEIS, arXiv:math/0409509 [math.CO], 2004.

Index entries for linear recurrences with constant coefficients, signature (4, -2, -8, 6, 6, -3, -2).

Formula

G.f.: x(2x^6 - 6x^5 + 8x^4 + 2x^3 - 6x^2 + 2x)/[(1 - x^2)(1 - 2x)(1 - x - x^2)^2] (conjectured). - Ralf Stephan, May 11 2004. Established by Burns and Purcell - see link.

a(0) = 1, a(1) = 0, a(2) = 2, a(3) = 2, a(4) = 6, a(5) = 12, a(6) = 26, a(7) = 58, a(n) = 4*a(n-1) - 2*a(n-2) - 8*a(n-3) + 6*a(n-4) + 6*a(n-5) - 3*a(n-6) - 2*a(n-7). - Harvey P. Dale, Sep 26 2012

a(n) = 2^n - 2 * n * Fibonacci(n-2) - (-1)^n - 1 for n >= 2 (proved by Burns and Purcell (2005, 2007)). - Petros Hadjicostas, Jul 04 2018

Example

11011001 is a winning string since 110{11}001 -> 11{000}1 -> {111} -> null.

Mathematica

Join[{1}, Rest[CoefficientList[Series[x (2x^6 - 6x^5 + 8x^4 + 2x^3 - 6x^2 + 2x)/((1 - x^2)(1 - 2x)(1 - x - x^2)^2), {x, 0, 40}], x]]] (* or *) Join[{1}, LinearRecurrence[{4, -2, -8, 6, 6, -3, -2}, {0, 2, 2, 6, 12, 26, 58}, 40]] (* Harvey P. Dale, Sep 26 2012 *)

Prog

(PARI) a(n)=if(n, ([0, 1, 0, 0, 0, 0, 0; 0, 0, 1, 0, 0, 0, 0; 0, 0, 0, 1, 0, 0, 0; 0, 0, 0, 0, 1, 0, 0; 0, 0, 0, 0, 0, 1, 0; 0, 0, 0, 0, 0, 0, 1; -2, -3, 6, 6, -8, -2, 4]^(n-1)*[0; 2; 2; 6; 12; 26; 58])[1, 1], 1) \\ Charles R Greathouse IV, Jun 15 2015

Crossrefs

Cf. A035617, A065237, A065238, A065239, A065240, A065241, A065242, A065243.

See A309874 for the losing strings.

For some similar questions in base 10, see A323830, A323831, A320487. - N. J. A. Sloane, Feb 04 2019

Row b=2 of A323844.

Sequence in context: A173392 A324128 A217211 * A115962 A019311 A216215

Adjacent sequences: A035612 A035613 A035614 * A035616 A035617 A035618

Keyword

nonn,nice,easy

Author

Erich Friedman

Extensions

More terms from Naohiro Nomoto, Jul 09 2001

Further terms from Sascha Kurz, Oct 19 2001

a(27)-a(36) from Robert Price, Apr 08 2019

Status

approved