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A196382 Number of sequences of n coin flips, that win on the last flip, if the sequence of flips ends with (1,1,0) or (1,0,1). 1
0, 0, 2, 3, 4, 7, 11, 16, 24, 36, 53, 78, 115, 169, 248, 364, 534, 783, 1148, 1683, 2467, 3616, 5300, 7768, 11385, 16686, 24455, 35841, 52528, 76984, 112826, 165355, 242340, 355167, 520523, 762864, 1118032, 1638556, 2401421, 3519454, 5158011 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

If the sequence ends with (1,1,0) Abel wins; if it ends with (1,0,1) Kain wins.

Abel(n)=A077868(n-3); Kain(n)=A000930(n-3).

Win probability for Abel=sum(Abel(n)/2^n)= 2/3.

Win probability for Kain=sum(Kain(n)/2^n)= 1/3.

Mean length of the game=sum(n*a(n)/2^n)= 6.

REFERENCES

A. Engel, Wahrscheinlichkeit und Statistik, Band 2, Klett, 1978, pages 25-26.

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-1).

FORMULA

a(n) = +2*a(n-1) -a(n-2) +a(n-3) -a(n-4), n>=5.

G.f.: x^3*(2-x)/((1-x)*(1-x-x^3)).

a(n) = 2*A077868(n-3) - A077868(n-4). - R. J. Mathar, Jan 11 2017

a(n) = a(n-1) + a(n-3) + 1, n>3. - Greg Dresden, Feb 09 2020

EXAMPLE

For n=6 the a(6)=7 solutions are (0,0,0,1,1,0),(1,0,0,1,1,0),(0,0,1,1,1,0),(0,1,1,1,1,0),(1,1,1,1,1,0) for Abel and (0,0,0,1,0,1),(1,0,0,1,0,1) for Kain.

MAPLE

a(1):=0: a(2):=0: a(3):=2: a(4):=3: a(5):=4:

for n from 6 to 100 do

  a(n):=a(n-1)+a(n-2)-a(n-5):

end do:

seq(a(n), n=1..100);

MATHEMATICA

Rest[CoefficientList[Series[x^3*(2 - x)/((1 - x)*(1 - x - x^3)), {x, 0, 50}], x]] (* G. C. Greubel, May 02 2017 *)

PROG

(PARI) x='x+O('x^50); concat([0, 0], Vec(x^3*(2 - x)/((1 - x)*(1 - x - x^3)))) \\ G. C. Greubel, May 02 2017

CROSSREFS

Cf. A000930, A077868, A179070 (first differences).

Sequence in context: A125621 A141001 A333260 * A120415 A023361 A210518

Adjacent sequences:  A196379 A196380 A196381 * A196383 A196384 A196385

KEYWORD

nonn,easy

AUTHOR

Paul Weisenhorn, Oct 28 2011

STATUS

approved

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Last modified July 27 11:04 EDT 2021. Contains 346304 sequences. (Running on oeis4.)