A065237 Number of winning length n strings with a 4-symbol alphabet in "same game".

1, 0, 4, 4, 28, 64, 268, 844, 3100, 10876, 39244, 142432, 518380, 1906012, 7012660, 25980940, 96407356, 359260936, 1341482740, 5023006444, 18844637356

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, Jul 05 2018

Links

Table of n, a(n) for n=0..20.

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]

Example

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

Crossrefs

Cf. A035615, A035617, A065238, A065239, A065240, A065241, A065242, A065243.

Row b=4 of A323844.

Sequence in context: A173049 A272040 A338672 * A264586 A348635 A307499

Adjacent sequences: A065234 A065235 A065236 * A065238 A065239 A065240

Keyword

nonn,more

Author

Sascha Kurz, Oct 23 2001

Extensions

a(13)-a(20) from Bert Dobbelaere, Dec 26 2018

Status

approved