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A050233 a(n) is the number of n-tosses having a run of 5 or more heads for a fair coin (i.e., probability is a(n)/2^n).

%I A050233 #45 Jun 05 2025 10:42:03
%S A050233 0,0,0,0,1,3,8,20,48,112,255,571,1262,2760,5984,12880,27553,58631,
%T A050233 124192,262008,550800,1154256,2412031,5027575,10455246,21697060,
%U A050233 44940472,92920992,191818561,395386763,813872712,1673157228,3435591712,7046697888,14438448127,29555251315,60444113566
%N A050233 a(n) is the number of n-tosses having a run of 5 or more heads for a fair coin (i.e., probability is a(n)/2^n).
%C A050233 a(n-1) is the number of compositions of n with at least one part >= 6. - _Joerg Arndt_, Aug 06 2012
%D A050233 W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 1, 2nd ed. New York: Wiley, p. 300, 1968.
%H A050233 T. D. Noe, <a href="/A050233/b050233.txt">Table of n, a(n) for n = 1..300</a>
%H A050233 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Run.html">Run</a>.
%H A050233 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-1,-1,-1,-2).
%F A050233 a(n) = 2^(n+1) - pentanacci(n+6), cf. A001591. - _Vladeta Jovovic_, Feb 23 2003
%F A050233 G.f.: x^5/((1-2*x)*(1-x-x^2-x^3-x^4-x^5)). - _Geoffrey Critzer_, Jan 29 2009
%F A050233 a(n) = 3*a(n-1) - a(n-2) - a(n-3) - a(n-4) - a(n-5) - 2*a(n-6). - _Wesley Ivan Hurt_, Jan 03 2021
%t A050233 f[x_] := x^4 / (1-3x+x^2+x^3+x^4+x^5+2x^6); CoefficientList[ Series[f[x], {x, 0, 31}], x] (* _Jean-François Alcover_, Nov 18 2011 *)
%t A050233 LinearRecurrence[{3,-1,-1,-1,-1,-2},{0,0,0,0,1,3},40] (* _Harvey P. Dale_, Jan 27 2015 *)
%o A050233 (PARI) a(n)=([0,1,0,0,0,0; 0,0,1,0,0,0; 0,0,0,1,0,0; 0,0,0,0,1,0; 0,0,0,0,0,1; -2,-1,-1,-1,-1,3]^(n-1)*[0;0;0;0;1;3])[1,1] \\ _Charles R Greathouse IV_, Jun 15 2015
%o A050233 (Magma)
%o A050233 R<x>:= PowerSeriesRing(Integers(), 50);
%o A050233 [0,0,0,0] cat Coefficients(R!( x^5/((1-2*x)*(1-x-x^2-x^3-x^4-x^5)) )); // _G. C. Greubel_, Jun 01 2025
%o A050233 (SageMath)
%o A050233 def A050233_list(prec):
%o A050233     P.<x>= PowerSeriesRing(QQ, prec)
%o A050233     return P( x^5/((1-2*x)*(1-x-x^2-x^3-x^4-x^5)) ).list()
%o A050233 a=A050233_list(41); a[1:] # _G. C. Greubel_, Jun 01 2025
%Y A050233 Cf. A001591, A008466, A050231, A050232.
%Y A050233 Column 6 of A109435.
%K A050233 nonn,nice,easy
%O A050233 1,6
%A A050233 _Eric W. Weisstein_