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A119706 Numerator of expected value of the longest run of heads in n tosses of a fair coin. The denominator is 2^n. 3
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, 2277559414, 4605810806, 9309804278, 18809961926 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

REFERENCES

A. M. Odlyzko, Asymptotic Enumeration Methods, pp. 136-137

R. Sedgewick and P. Flajolet, Analysis of Algorithms, Addison Wesley, 1996, page 372.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..1000

FORMULA

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

EXAMPLE

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.

MAPLE

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

# second Maple program:

b:= proc(n, m) option remember; `if`(n=0, 1,

      `if`(m=0, add(b(n-j, j), j=1..n),

      add(b(n-j, min(n-j, m)), j=1..min(n, m))))

    end:

a:= proc(n) option remember;

     `if`(n=1, 1, 2*a(n-1) +b(n, 0))

    end:

seq(a(n), n=1..40);  # Alois P. Heinz, Dec 19 2014

MATHEMATICA

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 *)

CROSSREFS

Sequence in context: A192965 A035593 A160399 * A034345 A036890 A000253

Adjacent sequences:  A119703 A119704 A119705 * A119707 A119708 A119709

KEYWORD

nonn

AUTHOR

Adam Kertesz (adamkertesz(AT)att.net), Jun 09 2006, Jun 13 2006

EXTENSIONS

More terms from R. J. Mathar, Jun 15 2006

STATUS

approved

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Last modified December 4 09:05 EST 2016. Contains 278749 sequences.