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A199925 Number of sequences of n coin flips that win on the last flip, if the sequence of flips ends with (0,0,1,1). 2
0, 0, 0, 1, 2, 4, 8, 14, 25, 45, 80, 142, 253, 450, 800, 1423, 2531, 4501, 8005, 14237, 25320, 45031, 80087, 142433, 253314, 450514, 801230, 1424971, 2534282, 4507169, 8015908, 14256129, 25354235, 45091990, 80195185, 142625502, 253656548 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

If the sequence ends with (1,1,1,1) Kain wins.

Kain(n)=A199926(n). Winsum(n)=A199594(n).

Win probability of Abel = sum(a(n)/2^n) = 3/4.

Win probability of Kain = sum(Kain(n)/2^n)=1/4.

Mean length of the game = sum(n*Winsum(n)/2^n)=12.

REFERENCES

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

LINKS

Table of n, a(n) for n=1..37.

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

FORMULA

a(n) = a(n-1)+a(n-2)+a(n-3)-a(n-5) for n>7.

a(n) = 2*a(n-1)-a(n-4)-a(n-5)+a(n-6) for n>8.

G.f.: x^4*(1+x)*(1+x^2)/(1-x-x^2-x^3+x^5).

a(n) = A164388(n-4) for n > 3. - Georg Fischer, Oct 14 2018

EXAMPLE

For n=6 the a(6)=4 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).

MAPLE

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

a(6):=4: a(7):=8: pot:=2^3: pa:=0:

for n from 4 to 7 do

pot:=2*pot:

pa:=pa+a(n)/pot:

end do:

for n from 8 to 100 do

pot:=2*pot:

a(n):=a(n-1)+a(n-2)+a(n-3)-a(n-5):

pa:=pa+a(n)/pot:

end do:

printf("%10.5f", pa):

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

MATHEMATICA

LinearRecurrence[{1, 1, 1, 0, -1}, {0, 0, 0, 1, 2, 4, 8}, 40] (* Harvey P. Dale, Jul 11 2018 *)

CROSSREFS

Cf. A164388, A199594, A199926.

Sequence in context: A164390 A164151 A281810 * A164388 A164389 A164401

Adjacent sequences: A199922 A199923 A199924 * A199926 A199927 A199928

KEYWORD

nonn

AUTHOR

Paul Weisenhorn, Nov 12 2011

STATUS

approved

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Last modified December 5 01:43 EST 2022. Contains 358572 sequences. (Running on oeis4.)