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

Author

Eric W. Weisstein

Keywords

• nonn
• nice
• easy

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
• 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.
• T. Langley, J. Liese, and J. 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).

Crossrefs

Cf. A000078, A008466, A050231, A050233.

Column 5 of A109435.

Programs

Magma

R:= 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

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 *)

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

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

SageMath

def A050232_list(prec):
    P.= 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

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