A065239 Number of winning length n strings with a 6-symbol alphabet in "same game".

1, 0, 6, 6, 66, 156, 966, 3366, 16986, 70386, 332646, 1484676, 6922146, 31921506, 149341506, 700002366, 3299514906, 15618721956, 74169285366

Offset

0,3

Comments

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

For binary strings, the formula for the number of winning strings of length n has been conjectured by Ralf Stephan and proved by Burns and Purcell (2005, 2007). For b-ary strings with b >= 3, the same problem seems to be unsolved. - Petros Hadjicostas, Aug 31 2019

Links

Table of n, a(n) for n=0..18.

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 in "same game", 2001. [ps file]

Sascha Kurz, Polynomials for same game, 2001. [pdf file]

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

Example

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

Crossrefs

Cf. A035615, A035617, A065237, A065238, A065240, A065241, A065242, A065243, A309874, A323812.

Row b=6 for A323844.

Sequence in context: A078290 A269888 A269767 * A146892 A347916 A361738

Adjacent sequences: A065236 A065237 A065238 * A065240 A065241 A065242

Keyword

nonn,more

Author

Sascha Kurz, Oct 23 2001

Extensions

a(12)-a(18) from Bert Dobbelaere, Dec 26 2018

Status

approved