A143286 Number of binary words of length n containing at least one subword 10^{6}1 and no subwords 10^{i}1 with i<6.

0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 9, 13, 19, 27, 37, 49, 63, 80, 102, 132, 173, 228, 300, 392, 508, 654, 839, 1076, 1382, 1778, 2289, 2945, 3783, 4850, 6207, 7934, 10135, 12943, 16526, 21095, 26915, 34320, 43733, 55692, 70882, 90174, 114673, 145778

Offset: 0

Author

Alois P. Heinz, Aug 04 2008

Keywords

• nonn
• easy

Examples

			a(9)=2 because 2 binary words of length 9 have at least one subword 10^{6}1 and no subwords 10^{i}1 with i<6: 010000001, 100000010.
		

Links

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

Crossrefs

Cf. A005709, A005710, 6th column of A143291.

Programs

Magma

[n le 8 select 0 else n le 15 select n-8 else 2*Self(n-1)-Self(n-2) +Self(n-7)-Self(n-9)-Self(n-15): n in [1..60]]; // Vincenzo Librandi, Jun 05 2013

Maple

a:= n-> coeff(series(x^8/((x^7+x-1)*(x^8+x-1)), x, n+1), x, n):
seq(a(n), n=0..60);

Mathematica

CoefficientList[Series[x^8 / ((x^7 + x - 1) (x^8 + x - 1)), {x, 0, 60}], x] (* Vincenzo Librandi, Jun 04 2013 *)

Formula

G.f.: x^8/((x^7+x-1)*(x^8+x-1)).

a(n) = A005709(n+6)-A005710(n+7).

a(n) = 2*a(n-1) - a(n-2) + a(n-7) - a(n-9) - a(n-15). - Vincenzo Librandi, Jun 05 2013