A143283 Number of binary words of length n containing at least one subword 10001 and no subwords 10^{i}1 with i<3.

0, 0, 0, 0, 0, 1, 2, 3, 4, 6, 10, 16, 24, 35, 51, 75, 110, 160, 231, 332, 476, 681, 972, 1384, 1966, 2787, 3944, 5573, 7864, 11082, 15597, 21926, 30791, 43199, 60553, 84807, 118682, 165966, 231930, 323905, 452083, 630629, 879224, 1225205, 1706532, 2375901

Offset

0,7

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,0,-1,0,0,-1).

Formula

G.f.: x^5/((x^4+x-1)*(x^5+x-1)).

a(n) = A003269(n+4) - A003520(n+4).

Example

a(9) = 6 because 6 binary words of length 8 have at least one subword 10001 and no subwords 11, 101 or 1001: 000010001, 000100010, 001000100, 010001000, 100010000, 100010001.

Maple

a:= proc(m) local M;
      M:= Matrix (2*m+3, (i, j)-> `if` (i=j-1 and i<>m+1 or j=1 and
                  i in [1, m+1] or j=m+2 and i in [m+2, 2*m+3], 1, 0));
      proc(n) local K; K:= M^(n+m+1); K[m+1, 1] -K[m+2, m+2] end
    end(3):
seq (a(n), n=0..55);

Prog

(PARI) Vec(x^5/((x^4+x-1)*(x^5+x-1))+O(x^99)) \\ Charles R Greathouse IV, May 30 2026

Crossrefs

Cf. A003269, A003520, 3rd column of A143291.

Sequence in context: A221996 A070542 A098855 * A104767 A072944 A024722

Adjacent sequences: A143280 A143281 A143282 * A143284 A143285 A143286

Keyword

nonn,easy

Author

Alois P. Heinz, Aug 04 2008

Status

approved