A065238 Number of winning length n strings with a 5-symbol alphabet in "same game".

1, 0, 5, 5, 45, 105, 545, 1825, 7965, 30845, 128945, 527785, 2202785, 9222985, 38818505, 164436125, 698347645, 2981306665, 12756855065

Offset: 0

Author

Sascha Kurz, Oct 23 2001

Keywords

• nonn
• more

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

Examples

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

Links

• 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.

Crossrefs

Cf. A035615, A035617, A065237, A065239, A065240, A065241, A065242, A065243, A309874, A323812.

Row b=5 of A323844.

Extensions

a(13)-a(18) from Bert Dobbelaere, Dec 26 2018