

A050231


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


13



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
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OFFSET

1,4


COMMENTS

a(n1) is the number of compositions of n with at least one part >=4.  Joerg Arndt, Aug 06 2012


REFERENCES

Feller, W. An Introduction to Probability Theory and Its Application, 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 [hepth], 2015.
Simon Cowell, A Formula for the Reliability of a ddimensional Consecutivekoutofn:F System, arXiv preprint arXiv:1506.03580 [math.CO], 2015.
Erich Friedman, Illustration of initial terms
T. Langley, J. Liese, 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).


FORMULA

a(n) = 2^n  tribonacci(n+3), see A000073.  Vladeta Jovovic, Feb 23 2003
G.f.: x^3/((12*x)*(1xx^2x^3)).  Geoffrey Critzer, Jan 29 2009
a(n) = 2 * a(n1) + 2^(n4)  a(n4) since we can add T or H to a sequence of n1 flips which has HHH, and H to one which ends in THH and does not have HHH among the first (n4) flips.  Toby Gottfried, Nov 20 2010
a(n) = 3*a(n1)  a(n2)  a(n3)  2*a(n4), a(0)=0, a(1)=0, a(2)=1, a(3)=3.  David Nacin, Mar 07 2012


MATHEMATICA

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


PROG

(Python)
def a(n, adict={0:0, 1:0, 2:1, 3:3}):
.if n in adict:
..return adict[n]
.adict[n]=3*a(n1)a(n2)a(n3)2*a(n4)
.return adict[n] # David Nacin, Mar 07 2012
(PARI) concat([0, 0], Vec(1/(12*x)/(1xx^2x^3)+O(x^99))) \\ Charles R Greathouse IV, Feb 03 2015


CROSSREFS

Cf. A000073, A008466, A050232, A050233.
Sequence in context: A006776 A291097 A293883 * A136305 A284943 A026712
Adjacent sequences: A050228 A050229 A050230 * A050232 A050233 A050234


KEYWORD

nonn,nice,easy


AUTHOR

Eric W. Weisstein


STATUS

approved



