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A171861 Expansion of x*(1+x+x^2) / ( (x-1)*(x^3+x^2-1) ). 21
1, 2, 4, 6, 9, 13, 18, 25, 34, 46, 62, 83, 111, 148, 197, 262, 348, 462, 613, 813, 1078, 1429, 1894, 2510, 3326, 4407, 5839, 7736, 10249, 13578, 17988, 23830, 31569, 41821, 55402, 73393, 97226, 128798, 170622, 226027, 299423, 396652 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Number of wins in Penney's game if the two players start HHT and TTT and HHT beats TTT.

HHT beats TTT 70% of the time. - Geoffrey Critzer, Mar 01 2014

LINKS

Table of n, a(n) for n=1..42.

Wikipedia, Penney's game

Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1).

FORMULA

a(n)= +a(n-1) +a(n-2) -a(n-4) = A000931(n+10)-3 = A134816(n+6)-3 = A078027(n+12)-3 .

EXAMPLE

a(n) enumerates length n+2 sequences on {H,T} that end in HHT but do not contain the contiguous subsequence TTT.

a(3)=4 because we have: TTHHT, THHHT, HTHHT, HHHHT.

a(4)=6 because we have: TTHHHT, THTHHT, THHHHT, HTTHHT, HTHHHT, HHHHHT. - Geoffrey Critzer, Mar 01 2014

MAPLE

A171861 := 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:

MATHEMATICA

nn=44; CoefficientList[Series[x(1+x+x^2)/(1-x-x^2+x^4), {x, 0, nn}], x] (* Geoffrey Critzer, Mar 01 2014 *)

PROG

(PARI) a(n)=([0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1; -1, 0, 1, 1]^(n-1)*[1; 2; 4; 6])[1, 1] \\ Charles R Greathouse IV, Oct 03 2016

CROSSREFS

Related sequences are A000045 (HHH beats HHT, HTT beats TTH), A006498 (HHH beats HTH), A023434 (HHH beats HTT), A000930 (HHH beats THT, HTH beats HHT), A000931 (HHH beats TTH), A077868 (HHT beats HTH), A002620 (HHT beats HTT), A000012 (HHT beats THH), A004277 (HHT beats THT), A070550 (HTH beats HHH), 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), A128588 (HTT beats TTT).

Sequence in context: A088575 A177189 A026906 * A039900 A039902 A081659

Adjacent sequences:  A171858 A171859 A171860 * A171862 A171863 A171864

KEYWORD

easy,nonn

AUTHOR

Ed Pegg Jr, Oct 16 2010

STATUS

approved

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Last modified December 5 16:08 EST 2016. Contains 278770 sequences.