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A050232
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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).
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8
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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
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OFFSET
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1,5
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COMMENTS
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a(n-1) is the number of compositions of n with at least one part >= 5. - Joerg Arndt, Aug 06 2012
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REFERENCES
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W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 1, 2nd ed. New York: Wiley, p. 300, 1968.
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LINKS
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Eric Weisstein's World of Mathematics, Run
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FORMULA
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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
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MATHEMATICA
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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 *)
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PROG
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(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)
(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
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CROSSREFS
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KEYWORD
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nonn,nice,easy
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AUTHOR
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STATUS
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approved
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