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).

0, 0, 0, 0, 1, 3, 8, 20, 48, 112, 255, 571, 1262, 2760, 5984, 12880, 27553, 58631, 124192, 262008, 550800, 1154256, 2412031, 5027575, 10455246, 21697060, 44940472, 92920992, 191818561, 395386763, 813872712, 1673157228, 3435591712, 7046697888, 14438448127, 29555251315, 60444113566

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn
• nice
• easy

Comments

a(n-1) is the number of compositions of n with at least one part >= 6. - Joerg Arndt, Aug 06 2012

References

• W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 1, 2nd ed. New York: Wiley, p. 300, 1968.

Links

• T. D. Noe, Table of n, a(n) for n = 1..300
• Eric Weisstein's World of Mathematics, Run.
• Index entries for linear recurrences with constant coefficients, signature (3,-1,-1,-1,-1,-2).

Crossrefs

Cf. A001591, A008466, A050231, A050232.

Column 6 of A109435.

Programs

Magma

R:= PowerSeriesRing(Integers(), 50);
[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

Mathematica

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 *)
LinearRecurrence[{3,-1,-1,-1,-1,-2},{0,0,0,0,1,3},40] (* Harvey P. Dale, Jan 27 2015 *)

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

SageMath

def A050233_list(prec):
    P.= PowerSeriesRing(QQ, prec)
    return P( x^5/((1-2*x)*(1-x-x^2-x^3-x^4-x^5)) ).list()
a=A050233_list(41); a[1:] # G. C. Greubel, Jun 01 2025

Formula

a(n) = 2^(n+1) - pentanacci(n+6), cf. A001591. - Vladeta Jovovic, Feb 23 2003

G.f.: x^5/((1-2*x)*(1-x-x^2-x^3-x^4-x^5)). - Geoffrey Critzer, Jan 29 2009

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