A143284 Number of binary words of length n containing at least one subword 100001 and no subwords 10^{i}1 with i<4.

0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 7, 11, 17, 25, 35, 48, 66, 92, 129, 180, 249, 342, 468, 640, 875, 1195, 1629, 2216, 3009, 4080, 5526, 7477, 10107, 13649, 18415, 24823, 33433, 44995, 60513, 81330, 109241, 146644, 196742, 263813, 353570, 473640, 634201

Offset: 0

Author

Alois P. Heinz, Aug 04 2008

Keywords

• nonn
• easy

Examples

			a(7)=2 because 2 binary words of length 7 have at least one subword 100001 and no subwords 10^{i}1 with i<4: 0100001, 1000010.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,1,0,-1,0,0,0,-1).

Crossrefs

Cf. A003520, A005708, 4th column of A143291.

Programs

Magma

[n le 6 select 0 else n le 11 select n-6 else 2*Self(n-1)-Self(n-2) +Self(n-5)-Self(n-7)-Self(n-11): n in [1..60]]; // Vincenzo Librandi, Jun 05 2013

Maple

a:= n-> coeff(series(x^6/((x^5+x-1)*(x^6+x-1)), x, n+1), x, n):
seq(a(n), n=0..60);

Mathematica

CoefficientList[Series[x^6 / ((x^5 + x - 1) (x^6 + x - 1)), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 04 2013 *)

Formula

G.f.: x^6/((x^5+x-1)*(x^6+x-1)).

a(n) = A003520(n+4) - A005708(n+5).

a(n) = 2*a(n-1)-a(n-2)+a(n-5)-a(n-7)-a(n-11). - Vincenzo Librandi, Jun 05 2013