A077868 -id:A077868 - cp's OEIS frontend

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).

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

Offset: 1

Paul Weisenhorn, Oct 28 2011

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.

			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.

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

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).

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

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);

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

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

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

A368299 a(n) is the number of permutations pi of [n] that avoid {231,321} so that pi^4 avoids 132.

Original entry on oeis.org

0, 1, 2, 4, 7, 13, 23, 41, 72, 127, 223, 392, 688, 1208, 2120, 3721, 6530, 11460, 20111, 35293, 61935, 108689, 190736, 334719, 587391, 1030800, 1808928, 3174448, 5570768, 9776017, 17155714, 30106180, 52832663, 92714861, 162703239, 285524281, 501060184, 879299327

Offset: 0

Kassie Archer, Dec 20 2023

Number of compositions of n of the form d_1+d_2+...+d_k=n where d_i is in {1,2,4} if i>1 and d_1 is any positive integer.

Alois P. Heinz, Table of n, a(n) for n = 0..2000
Kassie Archer and Aaron Geary, Powers of permutations that avoid chains of patterns, arXiv:2312.14351 [math.CO], 2023.
Index entries for linear recurrences with constant coefficients, signature (2,0,-1,1,-1).

Cf. A000071 (d_i in {1,2}), A077868 (d_i in {1,3}), A274110, A303666.

Partial sums of A181532.

a:= proc(n) option remember;
     `if`(n<1, 0, 1+add(a(n-j), j=[1, 2, 4]))
    end:
seq(a(n), n=0..37);  # Alois P. Heinz, Dec 20 2023

LinearRecurrence[{2,0,-1,1,-1},{0,1,2,4,7},38] (* Stefano Spezia, Dec 21 2023 *)

G.f.: x/((1-x)*(1-x-x^2-x^4)).
a(n) = Sum_{m=0..n-1} Sum_{r=0..floor(m/4)} Sum_{j=0..floor((m-4*r)/2)} binomial(m-3*r-j,r)*binomial(m-4*r-j,j).
a(n) = 1+a(n-1)+a(n-2)+a(n-4) where a(0)=0, a(1)=1, a(2)=2, a(3)=4.
a(n) = A274110(n+1) - 1.

cp's OEIS Frontend

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).

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