A151975 The number of ways one can flip seven consecutive tails (or heads) when flipping a coin n times.

0, 0, 0, 0, 0, 0, 0, 1, 3, 8, 20, 48, 112, 256, 576, 1279, 2811, 6126, 13256, 28512, 61008, 129952, 275712, 582913, 1228551, 2582048, 5412984, 11321744, 23631056, 49229312, 102377216, 212560127, 440668919, 912310222, 1886316324, 3895528632, 8035861664

Offset: 0

Author

Benjamin Merkel, Aug 05 2012

Keywords

• nonn
• easy

Comments

a(n-1) is the number of compositions of n with at least one part >=8. - Joerg Arndt, Aug 06 2012

Examples

			a(0)=0 means that there are no cases of seven consecutive tails (or heads) in zero coin flips.  Likewise, a(1)=a(2)=...=a(6)=0.  a(7)=1 since there is exactly one case of seven consecutive tails in seven coin flips.
		

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Benjamin E. Merkel, Probabilities of Consecutive Events in Coin Flipping, OhioLINK, 2011
• Index entries for linear recurrences with constant coefficients, signature (3,-1,-1,-1,-1,-1,-1,-2).

Crossrefs

Cf. A050231, A050232, A050233, A143662.

Programs

PARI

N=66;  x='x+O('x^N);
gf = (1-x)/(1-2*x); /* A011782(n): compositions of n */
gf -= 1/(1 - (x+x^2+x^3+x^4+x^5+x^6+x^7)); /* A066178(n): compositions of n into parts <=7 */
v151975=Vec(gf + 'a0);  v151975[1]=0; /* kludge to get all terms */
v151975 /* show terms */
/* Joerg Arndt, Aug 06 2012 */

PARI

concat(vector(7), Vec(x^7/((2*x-1)*(x^7+x^6+x^5+x^4+x^3+x^2+x-1)) + O(x^100))) \\ Colin Barker, Oct 16 2015

Formula

a(n) = A000079(n) - A066178(n+1).

G.f.: x^7 / ((2*x-1)*(x^7+x^6+x^5+x^4+x^3+x^2+x-1)). - Colin Barker, Oct 16 2015