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

0,12

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).

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A255377 A255378 A255379 * A255381 A078205 A316698

Adjacent sequences: A255377 A255378 A255379 * A255381 A255382 A255383

Keyword

nonn,base,easy

Author

Jon E. Schoenfield, Feb 22 2015

Status

approved