A050232 a(n) is the number of n-tosses having a run of 4 or more heads for a fair coin (i.e., probability is a(n)/2^n).

0, 0, 0, 1, 3, 8, 20, 48, 111, 251, 558, 1224, 2656, 5713, 12199, 25888, 54648, 114832, 240335, 501239, 1042126, 2160676, 4468664, 9221281, 18989899, 39034824, 80103276, 164126496, 335808927, 686182387, 1400438814, 2854992080, 5814293120, 11829648225, 24046855887, 48840756608

Offset

1,5

Comments

a(n-1) is the number of compositions of n with at least one part >= 5. - 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

Félix Balado and Guénolé C. M. Silvestre, Systematic Enumeration of Fundamental Quantities Involving Runs in Binary Strings, arXiv:2602.10005 [math.CO], 2026. See p. 20.

Simon Cowell, A Formula for the Reliability of a d-dimensional Consecutive-k-out-of-n:F System, arXiv preprint arXiv:1506.03580 [math.CO], 2015.

Thomas Langley, Jeffrey Liese, and Jeffrey Remmel, Generating Functions for Wilf Equivalence Under Generalized Factor Order, J. Int. Seq. 14 (2011) # 11.4.2.

Eric Weisstein's World of Mathematics, Run

Index entries for linear recurrences with constant coefficients, signature (3,-1,-1,-1,-2).

Formula

a(n) = 2^n - tetranacci(n+4), see A000078. - Vladeta Jovovic, Feb 23 2003

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

a(n) = 3*a(n-1) - a(n-2) - a(n-3) - a(n-4) - 2*a(n-5). - Wesley Ivan Hurt, Apr 23 2021

Mathematica

Flatten[With[{tetrnos=LinearRecurrence[{1, 1, 1, 1}, {0, 1, 1, 2}, 50]}, Table[ 2^n- Take[tetrnos, {n+3}], {n, 40}]]] (* Harvey P. Dale, Dec 02 2011 *)
LinearRecurrence[{3, -1, -1, -1, -2}, {0, 0, 0, 1, 3}, 31] (* Ray Chandler, Aug 03 2015 *)

Prog

(Python)
def a(n, adict={0:0, 1:0, 2:0, 3:1, 4:3}):
    if n in adict:
        return adict[n]
    adict[n]=3*a(n-1) - a(n-2) - a(n-3) - a(n-4) - 2*a(n-5)
    return adict[n] # David Nacin, Mar 07 2012
(PARI) a(n)=([0, 1, 0, 0, 0; 0, 0, 1, 0, 0; 0, 0, 0, 1, 0; 0, 0, 0, 0, 1; -2, -1, -1, -1, 3]^(n-1)*[0; 0; 0; 1; 3])[1, 1] \\ Charles R Greathouse IV, Feb 09 2017
(Magma)
R<x>:= PowerSeriesRing(Integers(), 50);
[0, 0, 0] cat Coefficients(R!( x^4/((1-2*x)*(1-x-x^2-x^3-x^4)) )); // G. C. Greubel, Jun 01 2025
(SageMath)
def A050232_list(prec):
    P.<x>= PowerSeriesRing(QQ, prec)
    return P( x^4/((1-2*x)*(1-x-x^2-x^3-x^4)) ).list()
a=A050232_list(41); a[1:] # G. C. Greubel, Jun 01 2025

Crossrefs

Cf. A000078, A008466, A050231, A050233.

Column 5 of A109435.

Sequence in context: A284943 A026712 A308370 * A050233 A143662 A151975

Adjacent sequences: A050229 A050230 A050231 * A050233 A050234 A050235

Keyword

nonn,nice,easy

Author

Eric W. Weisstein

Status

approved