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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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%I #67 May 22 2024 02:11:27

%S 0,0,1,3,8,20,47,107,238,520,1121,2391,5056,10616,22159,46023,95182,

%T 196132,402873,825259,1686408,3438828,6999071,14221459,28853662,

%U 58462800,118315137,239186031,483072832,974791728,1965486047

%N 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).

%C a(n-1) is the number of compositions of n with at least one part >= 4. - _Joerg Arndt_, Aug 06 2012

%D W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 1, 2nd ed. New York: Wiley, p. 300, 1968.

%H T. D. Noe, <a href="/A050231/b050231.txt">Table of n, a(n) for n = 1..300</a>

%H David Broadhurst, <a href="http://arxiv.org/abs/1504.05303">Multiple Landen values and the tribonacci numbers</a>, arXiv:1504.05303 [hep-th], 2015.

%H Simon Cowell, <a href="http://arxiv.org/abs/1506.03580">A Formula for the Reliability of a d-dimensional Consecutive-k-out-of-n:F System</a>, arXiv preprint arXiv:1506.03580 [math.CO], 2015.

%H Erich Friedman, <a href="/A050231/a050231.gif">Illustration of initial terms</a>

%H T. Langley, J. Liese, and J. Remmel, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Langley/langley2.html">Generating Functions for Wilf Equivalence Under Generalized Factor Order</a>, J. Int. Seq. 14 (2011) # 11.4.2.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Run.html">Run</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-1,-2).

%F a(n) = 2^n - tribonacci(n+3), see A000073. - _Vladeta Jovovic_, Feb 23 2003

%F G.f.: x^3/((1-2*x)*(1-x-x^2-x^3)). - _Geoffrey Critzer_, Jan 29 2009

%F 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

%F 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

%t LinearRecurrence[{3, -1, -1, -2}, {0, 0, 1, 3}, 50] (* _David Nacin_, Mar 07 2012 *)

%o (Python)

%o def a(n, adict={0:0, 1:0, 2:1, 3:3}):

%o if n in adict:

%o return adict[n]

%o adict[n]=3*a(n-1)-a(n-2)-a(n-3)-2*a(n-4)

%o return adict[n] # _David Nacin_, Mar 07 2012

%o (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

%Y Cf. A000073, A008466, A050232, A050233.

%K nonn,nice,easy

%O 1,4

%A _Eric W. Weisstein_