OFFSET
2,2
COMMENTS
For n >= 2, a(n) is one-half the number of length n losing strings with a binary alphabet in the "same game".
In the "same game", winning strings are those that can be reduced to the null string by repeatedly removing an entire run of two or more consecutive symbols.
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 for same game, pdf.
Ralf Stephan, Prove or disprove: 100 conjectures from the OEIS, arXiv:math/0409509 [math.CO], 2004.
FORMULA
a(n) = A309874(n)/2 for n >= 2.
EXAMPLE
11011001 is a winning string because 110{11}001 -> 11{000}1 -> {111} -> null. Its complement, 00100110 is also a winning string because 001{00}110 -> 00{111}0 -> {000} -> null.
MATHEMATICA
Table[n Fibonacci[n-2]+((-1)^n+1)/2, {n, 2, 40}] (* Harvey P. Dale, Sep 17 2019 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Petros Hadjicostas, Sep 01 2019
STATUS
approved