cp's OEIS Frontend

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

%I A151975 #40 Jan 02 2019 16:25:46
%S A151975 0,0,0,0,0,0,0,1,3,8,20,48,112,256,576,1279,2811,6126,13256,28512,
%T A151975 61008,129952,275712,582913,1228551,2582048,5412984,11321744,23631056,
%U A151975 49229312,102377216,212560127,440668919,912310222,1886316324,3895528632,8035861664
%N A151975 The number of ways one can flip seven consecutive tails (or heads) when flipping a coin n times.
%C A151975 a(n-1) is the number of compositions of n with at least one part >=8. - _Joerg Arndt_, Aug 06 2012
%H A151975 Colin Barker, <a href="/A151975/b151975.txt">Table of n, a(n) for n = 0..1000</a>
%H A151975 Benjamin E. Merkel, <a href="http://rave.ohiolink.edu/etdc/view?acc_num=ucin1307442290">Probabilities of Consecutive Events in Coin Flipping</a>, OhioLINK, 2011
%H A151975 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-1,-1,-1,-1,-1,-2).
%F A151975 a(n) = A000079(n) - A066178(n+1).
%F A151975 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
%e A151975 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.
%o A151975 (PARI) N=66;  x='x+O('x^N);
%o A151975 gf = (1-x)/(1-2*x); /* A011782(n): compositions of n */
%o A151975 gf -= 1/(1 - (x+x^2+x^3+x^4+x^5+x^6+x^7)); /* A066178(n): compositions of n into parts <=7 */
%o A151975 v151975=Vec(gf + 'a0);  v151975[1]=0; /* kludge to get all terms */
%o A151975 v151975 /* show terms */
%o A151975 /* _Joerg Arndt_, Aug 06 2012 */
%o A151975 (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
%Y A151975 Cf. A050231, A050232, A050233, A143662.
%K A151975 nonn,easy
%O A151975 0,9
%A A151975 _Benjamin Merkel_, Aug 05 2012