A143288 Number of binary words of length n containing at least one subword 10^{8}1 and no subwords 10^{i}1 with i<8.
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 15, 21, 29, 39, 51, 65, 81, 99, 120, 146, 180, 225, 284, 360, 456, 575, 720, 895, 1106, 1362, 1676, 2065, 2550, 3156, 3912, 4851, 6011, 7437, 9184, 11321, 13936, 17141, 21077, 25919, 31881, 39222, 48254
Offset: 0
Examples
a(11)=2 because 2 binary words of length 11 have at least one subword 10^{8}1 and no subwords 10^{i}1 with i<8: 01000000001, 10000000010.
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,-1).
Programs
-
Maple
a:= n-> coeff(series(x^10/((x^9+x-1)*(x^10+x-1)), x, n+1), x, n): seq(a(n), n=0..70);
-
Mathematica
CoefficientList[Series[x^10 / ((x^9 + x - 1) (x^10 + x - 1)), {x, 0, 60}], x] (* Vincenzo Librandi, Jun 04 2013 *) LinearRecurrence[{2,-1,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,-1},{0,0,0,0,0,0,0,0,0,0,1,2,3,4,5,6,7,8,9},60] (* Harvey P. Dale, Oct 12 2018 *) -
PARI
Vec(1/((x^9+x-1)(x^10+x-1))+O(x^99)) \\ Charles R Greathouse IV, Jun 04 2013
Formula
G.f.: x^10/((x^9+x-1)*(x^10+x-1)).
a(n) = 2a(n-1) - a(n-2) + a(n-9) - a(n-11) - a(n-19). - Charles R Greathouse IV, Jun 04 2013