A143290 Number of binary words of length n containing at least one subword 10^{10}1 and no subwords 10^{i}1 with i<10.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 17, 23, 31, 41, 53, 67, 83, 101, 121, 143, 168, 198, 236, 285, 348, 428, 528, 651, 800, 978, 1188, 1434, 1722, 2061, 2464, 2948, 3534, 4247, 5116, 6174, 7458, 9009, 10873, 13103, 15762, 18927

Offset: 0

Author

Alois P. Heinz, Aug 04 2008

Keywords

• nonn
• easy

Examples

			a(13)=2 because 2 binary words of length 13 have at least one subword 10^{10}1 and no subwords 10^{i}1 with i<10: 0100000000001, 1000000000010.
		

Links

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

Crossrefs

Cf. A017905, A017906, 10th column of A143291.

Programs

Magma

[n le 12 select 0 else n le 23 select n-12 else 2*Self(n-1)-Self(n-2) +Self(n-11)-Self(n-13)-Self(n-23): n in [1..60]]; // Vincenzo Librandi, Jun 05 2013

Maple

a:= n-> coeff(series(x^12/((x^11+x-1)*(x^12+x-1)), x, n+1), x, n):
seq(a(n), n=0..60);

Mathematica

CoefficientList[Series[x^12 / ((x^11 + x - 1) (x^12 + x - 1)), {x, 0, 60}], x] (* Vincenzo Librandi, Jun 05 2013 *)
LinearRecurrence[{2,-1,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,-1},{0,0,0,0,0,0,0,0,0,0,0,0,1,2,3,4,5,6,7,8,9,10,11},80] (* Harvey P. Dale, Aug 20 2021 *)

Formula

G.f.: x^12/((x^11+x-1)*(x^12+x-1)).

a(n) = A017905(n+21)-A017906(n+23).

a(n) = 2*a(n-1) -a(n-2) +a(n-11) -a(n-13) -a(n-23). - Vincenzo Librandi, Jun 05 2013