# [seqfan] Re: Penney's game sequences

Richard Mathar mathar at strw.leidenuniv.nl
Sat Oct 16 16:03:42 CEST 2010

```http://list.seqfan.eu/pipermail/seqfan/2010-October/006234.html says

epj> {1,2,4,6,9,13,18,25,34,46,62,83,111,148,197,262,348}         (*HHT beats TTT   Not in OEIS*),

Every sequence is implicitly in the OEIS :-). This here is apparently

%I A000001
%S A000001 1,2,4,6,9,13,18,25,34,46,62,83,111,148,197,262,348,462,613,813,
%T A000001 1078,1429,1894,2510,3326,4407,5839,7736,10249,13578,17988,23830,
%U A000001 31569,41821,55402,73393,97226,128798,170622,226027,299423,396652
%N A000001 Expansion of x*(1+x+x^2) / ( (x-1)*(x^3+x^2-1) ).
%C A000001 Number of wins in Penney's game if the two players start HHT and TTT and HHT beats TTT.
%C A000001 Related sequences are A000045 (HHH beats HHT),
A006498 (HHH beats HTH),
A023434 (HHH beats HTT),
A000930 (HHH beats THT),
A000931 (HHH beats TTH),
A077868 (HHT beats HTH),
A002620 (HHT beats HTT),
A000012 (HHT beats THH),
A004277 (HHT beats THT),
A070550 (HTH beats HHH),
A000930 (HTH beats HHT),
A000027 (HTH beats HTT),
A097333 (HTH beats THH),
A040000 (HTH beats TTH),
A068921 (HTH beats TTT),
A054405 (HTT beats HHH),
A008619 (HTT beats HHT),
A038718 (HTT beats THT),
A000045 (HTT beats TTH),
A128588 (HTT beats TTT).
%H A000001 Anonymous, <a href="http://en.wikipedia.org/wiki/Penney's_game">Penney's game</a>, Wikipedia
%F A000001 a(n)= +a(n-1) +a(n-2) -a(n-4) = A000931(n+10)-3 = A134816(n+6)-3 = A078027(n+12)-3 .
%e A000001
%p A000001 A000001 := proc(n) option remember; if n <=4 then op(n,[1,2,4,6]); else procname(n-1)+procname(n-2)-procname(n-4) ; end if; end proc:
%Y A000001
%K A000001 easy,nonn
%O A000001 1,2
%A A000001 Ed Pegg Jr (ed(AT)mathpuzzle.com), Oct 16 2010

```