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A119706
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Numerator of expected value of the longest run of heads in n tosses of a fair coin. The denominator is 2^n.
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3
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1, 4, 11, 27, 62, 138, 300, 643, 1363, 2866, 5988, 12448, 25770, 53168, 109381, 224481, 459742, 939872, 1918418, 3910398, 7961064, 16190194, 32893738, 66772387, 135437649, 274518868, 556061298, 1125679616
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OFFSET
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1,2
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COMMENTS
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a(n) is also the sum of the number of binary words with at least one run of consecutive 0's of length >= i for i>=1. In other words A000225 + A008466 + A050231 + A050232 + ... - Geoffrey Critzer, Jan 12 2013
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REFERENCES
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A. M. Odlyzko, Asymptotic Enumeration Methods, pp. 136-137
R. Sedgewick and P. Flajolet, Analysis of Algorithms, Addison Wesley, 1996, page 372.
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LINKS
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Table of n, a(n) for n=1..28.
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FORMULA
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a(n+1) = 2*a(n) + A007059(n+2)
a(n) > 2*a(n-1). a(n)=sum(i=1..(2^n)-1, A038374(i) ). - R. J. Mathar, Jun 15 2006
O.g.f.: sum(k>=1, 1/(1-2*x) - (1-x^k)/(1-2*x-x^(k+1)) ).
a(n) = sum(k=1..n, A048004(n,k) * k ). - Geoffrey Critzer, Jan 12 2013
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EXAMPLE
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a(3)=11 because for the 8(2^3) possible runs 0 is longest run of heads once, 1 four times, 2 two times and 3 once and 0*1+1*4+2*2+3*1=11
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MAPLE
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A038374 := proc(n) local nshft, thisr, resul; nshft := n ; resul :=0 ; thisr :=0 ; while nshft > 0 do if nshft mod 2 <> 0 then thisr := thisr+1 ; else resul := max(resul, thisr) ; thisr := 0 ; fi ; nshft := floor(nshft/2) ; od ; resul := max(resul, thisr) ; RETURN(resul) ; end : A119706 := proc(n) local count, c, rlen ; count := array(0..n) ; for c from 0 to n do count[c] := 0 ; od ; for c from 0 to 2^n-1 do rlen := A038374(c) ; count[rlen] := count[rlen]+1 ; od ; RETURN( sum('count[c]*c', 'c'=0..n) ); end: for n from 1 to 40 do print(n, A119706(n)) ; od : - R. J. Mathar, Jun 15 2006
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MATHEMATICA
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nn=10; Drop[Apply[Plus, Table[CoefficientList[Series[1/(1-2x)-(1-x^n)/(1-2x+x^(n+1)), {x, 0, nn}], x], {n, 1, nn}]], 1] (* Geoffrey Critzer, Jan 12 2013 *)
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CROSSREFS
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Sequence in context: A192965 A035593 A160399 * A034345 A036890 A000253
Adjacent sequences: A119703 A119704 A119705 * A119707 A119708 A119709
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KEYWORD
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nonn
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AUTHOR
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Adam Kertesz (adamkertesz(AT)att.net), Jun 09 2006, Jun 13 2006
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EXTENSIONS
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More terms from R. J. Mathar, Jun 15 2006
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STATUS
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approved
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