A050231 a(n) is the number of n-tosses having a run of 3 or more heads for a fair coin (i.e., probability is a(n)/2^n).

0, 0, 1, 3, 8, 20, 47, 107, 238, 520, 1121, 2391, 5056, 10616, 22159, 46023, 95182, 196132, 402873, 825259, 1686408, 3438828, 6999071, 14221459, 28853662, 58462800, 118315137, 239186031, 483072832, 974791728, 1965486047, 3960221519, 7974241118, 16047432332, 32276862265

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 >= 4. - 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
• David Broadhurst, Multiple Landen values and the tribonacci numbers, arXiv:1504.05303 [hep-th], 2015.
• 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.
• Erich Friedman, Illustration of initial terms
• 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,-2).

Crossrefs

Cf. A000073, A008466, A050232, A050233.

Column 4 of A109435.

Programs

Magma

R:= PowerSeriesRing(Integers(), 60);
[0,0] cat Coefficients(R!( x^3/((1-2*x)*(1-x-x^2-x^3)) )); // G. C. Greubel, Jun 01 2025

Mathematica

LinearRecurrence[{3, -1, -1, -2}, {0, 0, 1, 3}, 50] (* David Nacin, Mar 07 2012 *)

PARI

concat([0,0], Vec(1/(1-2*x)/(1-x-x^2-x^3)+O(x^99))) \\ Charles R Greathouse IV, Feb 03 2015

Python

def a(n, adict={0:0, 1:0, 2:1, 3:3}):
    if n in adict:
        return adict[n]
    adict[n]=3*a(n-1)-a(n-2)-a(n-3)-2*a(n-4)
    return adict[n] # David Nacin, Mar 07 2012

SageMath

def A050231_list(prec):
    P.= PowerSeriesRing(QQ, prec)
    return P( x^3/((1-2*x)*(1-x-x^2-x^3)) ).list()
a=A050231_list(41); a[1:] # G. C. Greubel, Jun 01 2025

Formula

a(n) = 2^n - tribonacci(n+3), see A000073. - Vladeta Jovovic, Feb 23 2003

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

a(n) = 2 * a(n-1) + 2^(n-4) - a(n-4) since we can add T or H to a sequence of n-1 flips which has HHH, and H to one which ends in THH and does not have HHH among the first (n-4) flips. - Toby Gottfried, Nov 20 2010

a(n) = 3*a(n-1) - a(n-2) - a(n-3) - 2*a(n-4), a(0)=0, a(1)=0, a(2)=1, a(3)=3. - David Nacin, Mar 07 2012