A143285 Number of binary words of length n containing at least one subword 1000001 and no subwords 10^{i}1 with i<5.

0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 8, 12, 18, 26, 36, 48, 63, 83, 111, 150, 203, 273, 364, 482, 636, 839, 1108, 1464, 1933, 2548, 3352, 4402, 5774, 7568, 9914, 12980, 16983, 22204, 29008, 37870, 49408, 64425, 83963, 109373, 142406, 185331, 241088, 313486

Offset: 0

Author

Alois P. Heinz, Aug 04 2008

Keywords

• nonn
• easy

Examples

			a(8)=2 because 2 binary words of length 8 have at least one subword 1000001 and no subwords 10^{i}1 with i<5: 01000001, 10000010.
		

Links

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

Crossrefs

Cf. A005708, A005709, 5th column of A143291.

Programs

Magma

[n le 7 select 0 else n le 13 select n-7 else 2*Self(n-1)-Self(n-2) +Self(n-6)-Self(n-8)-Self(n-13): n in [1..60]]; // Vincenzo Librandi, Jun 05 2013

Maple

a:= n-> coeff(series(x^7/((x^6+x-1)*(x^7+x-1)), x, n+1), x, n):
seq(a(n), n=0..60);

Mathematica

CoefficientList[Series[x^7 / ((x^6 + x - 1) (x^7 + x - 1)), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 04 2013 *)

Formula

G.f.: x^7/((x^6+x-1)*(x^7+x-1)).

a(n) = A005708(n+5) - A005709(n+6).

a(n) = 2*a(n-1) -a(n-2) +a(n-6) -a(n-8) -a(n-13). - Vincenzo Librandi, Jun 05 2013