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A050231
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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).
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13
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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
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1,4
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COMMENTS
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a(n-1) is the number of compositions of n with at least one part >=4. - Joerg Arndt, Aug 06 2012
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REFERENCES
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Feller, W. An Introduction to Probability Theory and Its Application, Vol. 1, 2nd ed. New York: Wiley, p. 300, 1968.
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LINKS
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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, 2015
Erich Friedman, Illustration of initial terms
Eric Weisstein's World of Mathematics, Run
Index entries for linear recurrences with constant coefficients, signature (3,-1,-1,-2).
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FORMULA
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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
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MATHEMATICA
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LinearRecurrence[{3, -1, -1, -2}, {0, 0, 1, 3}, 50] (* David Nacin, Mar 07 2012 *)
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PROG
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(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
(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
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CROSSREFS
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Cf. A008466, A050232, A050233, A000073.
Sequence in context: A138803 A048492 A006776 * A136305 A284943 A026712
Adjacent sequences: A050228 A050229 A050230 * A050232 A050233 A050234
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KEYWORD
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nonn,nice,easy
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AUTHOR
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Eric W. Weisstein
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STATUS
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approved
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