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A164139 Number of binary strings of length n with equal numbers of 000 and 011 substrings. 1
1, 2, 4, 6, 9, 14, 22, 38, 66, 120, 222, 416, 792, 1512, 2909, 5610, 10851, 21042, 40864, 79514, 154911, 302210, 590251, 1154012, 2258488, 4423856, 8672541, 17014530, 33404100, 65624480, 129002143, 253733246, 499333096, 983154996, 1936685718 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

R. H. Hardin and Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 501 terms from R. H. Hardin)

Shalosh B. Ekhad and Doron Zeilberger, Automatic Solution of Richard Stanley's Amer. Math. Monthly Problem #11610 and ANY Problem of That Type, arXiv preprint arXiv:1112.6207 [math.CO], 2011. See subpages for rigorous derivations of g.f., recurrence, asymptotics for this sequence. [From N. J. A. Sloane, Apr 07 2012]

MAPLE

a:= proc(n) a(n):= `if`(n<8, [1, 2, 4, 6, 9, 14, 22, 38][n+1],

       ((32*n^2-136*n+18)*a(n-1) +(8*n^2-66*n+52)*a(n-2)

        -(46*n^2-225*n+11)*a(n-3) -2*(n-3)*(4*n-25)*a(n-4)

        +(72*n^2-448*n+400)*a(n-5) -(69*n^2-445*n+464)*a(n-6)

        -(4*n^2-54*n-78)*a(n-7) +4*(7*n-4)*(n-6)*a(n-8))/

        (n*(13*n-56)))

    end:

seq(a(n), n=0..50);  # Alois P. Heinz, Aug 29 2014

MATHEMATICA

a[n_] := If[n < 8, {1, 2, 4, 6, 9, 14, 22, 38}[[n + 1]], (1/(n*(13*n - 56)))*(a*(-(4*n^2 - 54*n - 78))*(n - 7) - (69*n^2 - 445*n + 464)*a[n - 6] + (72*n^2 - 448*n + 400)*a[n - 5] - (46*n^2 - 225*n + 11)*a[n - 3] + (8*n^2 - 66*n + 52)*a[n - 2] + (32*n^2 - 136*n + 18)*a[n - 1] + 4*(n - 6)*(7*n - 4)*a[n - 8] - 2*(n - 3)*(4*n - 25)*a[n - 4])];

Table[a[n], {n, 0, 50}] (* Jean-Fran├žois Alcover, Nov 11 2017, after Alois P. Heinz *)

CROSSREFS

Sequence in context: A042942 A256968 A005687 * A218605 A024849 A323227

Adjacent sequences:  A164136 A164137 A164138 * A164140 A164141 A164142

KEYWORD

nonn

AUTHOR

R. H. Hardin, Aug 11 2009

STATUS

approved

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Last modified September 19 18:06 EDT 2020. Contains 337181 sequences. (Running on oeis4.)