A035617 Number of winning length n strings with a 3-symbol alphabet in "same game".

1, 0, 3, 3, 15, 33, 105, 297, 879, 2631, 7833, 23697, 71385, 216765, 657849, 2003151, 6103743, 18624693, 56870385, 173760513, 531128349, 1623881889, 4965695331, 15185222199, 46435889601, 141985777503

Offset: 0

Author

Erich Friedman

Keywords

• nonn
• nice
• 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, Dec 27 2018

Examples

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

Links

• C. Burns and B. Purcell, A note on Stephan's conjecture 77, preprint, 2005.
• C. Burns and B. 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 in "same game", 2001. [pdf file]

Crossrefs

Cf. A035615, A065237, A065238, A065239, A065240, A065241, A065242, A065243.

Row b=3 of A323844.

Extensions

a(16)-a(25) from Bert Dobbelaere, Dec 26 2018