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%I #38 Feb 19 2023 17:30:47
%S 0,0,2,2,4,6,10,16,26,42,68,110,178,288,466,754,1220,1974,3194,5168,
%T 8362,13530,21892,35422,57314,92736,150050,242786,392836,635622,
%U 1028458,1664080,2692538,4356618,7049156,11405774,18454930,29860704,48315634,78176338
%N Number of sequences of n coin flips that win on the last flip, if the sequence of flips ends with (0,1,1) or (1,1,1).
%C If the sequence ends with (011) Abel wins; if it ends with (111) Kain wins.
%C Kain(n)=0 for n <> 3; Kain(3)=1.
%C Abel(n) = A128588(n-2) for n > 2.
%C a(n) = A006355(n-1) for n > 2.
%C Win probability for Abel: Sum_{n>=1} Abel(n)/2^n = 7/8.
%C Win probability for Kain: Kain(3)/8 = 1/8.
%C Mean length of the game: Sum_{n>=1} n*a(n)/2^n = 7.
%C Appears to be essentially the same as A163733, A118658, A055389. - _R. J. Mathar_, Oct 31 2011
%D A. Engel, Wahrscheinlichkeit und Statistik, Band 2, Klett, 1978, pages 25-26.
%H Vincenzo Librandi, <a href="/A198834/b198834.txt">Table of n, a(n) for n = 1..1000</a>
%H Tian-Xiao He, Peter J.-S. Shiue, Zihan Nie, Minghao Chen, <a href="https://doi.org/10.3934/era.2020057">Recursive sequences and Girard-Waring identities with applications in sequence transformation</a>, Electronic Research Archive (2020) Vol. 28, No. 2, 1049-1062.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,1).
%F a(n) = a(n-1) + a(n-2) for n > 3.
%F G.f.: 2*x^3/(1 - x - x^2).
%F a(n) = 2*A000045(n-2). - _R. J. Mathar_, Jan 11 2017
%F E.g.f.: 2 - 2*x + 2*exp(x/2)*(3*sqrt(5)*sinh(sqrt(5)*x/2) - 5*cosh(sqrt(5)*x/2))/5. - _Stefano Spezia_, Feb 19 2023
%e For n=6 the a(6)=6 solutions are (0,0,0,0,1,1), (1,0,0,0,1,1); (0,1,0,0,1,1), (1,1,0,0,1,1), (0,0,1,0,1,1), (1,0,1,0,1,1) all for Abel.
%p a(1):=0: a(2):=0: a(3):=2:
%p ml:=0.75: pot:=8:
%p for n from 4 to 100 do
%p pot:=2*pot:
%p a(n):=a(n-1)+a(n-2):
%p ml:=ml+n*a(n)/pot:
%p end do:
%p printf("%12.8f",ml);
%p seq(a(n),n=1..100);
%t Join[{0, 0}, Table[2*Fibonacci[n], {n, 70}]] (* _Vladimir Joseph Stephan Orlovsky_, Feb 10 2012 *)
%t Join[{0},LinearRecurrence[{1,1},{0,2},50]] (* _Vincenzo Librandi_, Feb 19 2012 *)
%Y Cf. A006355, A055389, A118658, A128588, A163733.
%K nonn,easy
%O 1,3
%A _Paul Weisenhorn_, Oct 30 2011
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