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

Offset

1,6

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).

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

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 *)

Prog

(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

Crossrefs

Cf. A001591, A008466, A050231, A050232.

Sequence in context: A026712 A308370 A050232 * A143662 A151975 A168150

Adjacent sequences: A050230 A050231 A050232 * A050234 A050235 A050236

Keyword

nonn,nice,easy

Author

Eric W. Weisstein

Status

approved