A255380 Number of strings of n decimal digits that contain at least one string of exactly 10 consecutive "0" digits.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 18, 261, 3420, 42300, 504000, 5850000, 66600000, 747000000, 8280000000, 90900000000, 989999999991, 10709999999757, 115199999995383, 1232999999925399, 13139999998904070, 139499999984885400, 1475999999800740000

Offset: 0

Author

Jon E. Schoenfield, Feb 22 2015

Keywords

• nonn
• base
• easy

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (20,-100,0,0,0,0,0,0,0,0,-9,99,-90).

Crossrefs

Cf. A255371-A255379 (for the k=1 through k=9 cases of "exactly k consecutive '0' digits").

Programs

Mathematica

CoefficientList[Series[x^10(x-1)^2/((10x-1)(9x^12-9x^11+10x-1)),{x,0,40}],x] (* or *) LinearRecurrence[{20,-100,0,0,0,0,0,0,0,0,-9,99,-90},{0,0,0,0,0,0,0,0,0,0,1,18,261},40] (* Harvey P. Dale, Dec 27 2021 *)

PARI

concat([0,0,0,0,0,0,0,0,0,0], Vec(x^10*(x-1)^2/((10*x-1)*(9*x^12-9*x^11+10*x-1)) + O(x^100))) \\ Colin Barker, Feb 27 2015

Formula

a(0) = a(1) = ... = a(9) = 0, a(10) = 1, a(n) = 9*(10^(n-11) - a(n-11) + sum_{i=10..n-1} a(i)) for n>=11.

G.f.: x^10*(x-1)^2/((10*x-1)*(9*x^12-9*x^11+10*x-1)). - Alois P. Heinz, Feb 26 2015