A255378 Number of strings of n decimal digits that contain at least one string of exactly 8 consecutive "0" digits.

0, 0, 0, 0, 0, 0, 0, 0, 1, 18, 261, 3420, 42300, 504000, 5850000, 66600000, 747000000, 8279999991, 90899999757, 989999995383, 10709999925399, 115199998904070, 1232999984885400, 13139999800740000, 139499997460650000, 1475999968482900000, 15569999617032000081

Offset: 0

Author

Jon E. Schoenfield, Feb 22 2015

Keywords

• nonn
• base
• easy

Links

• Index entries for linear recurrences with constant coefficients, signature (20,-100,0,0,0,0,0,0,-9,99,-90).

Crossrefs

Cf. A255371-A255377 (for the k=1 through k=7 cases of "exactly k consecutive '0' digits"), A255379 and A255380 (for the cases k=9 and k=10).

Programs

Mathematica

LinearRecurrence[{20,-100,0,0,0,0,0,0,-9,99,-90},{0,0,0,0,0,0,0,0,1,18,261},30] (* Harvey P. Dale, Mar 26 2022 *)

Formula

a(0) = a(1) = ... = a(7) = 0, a(8) = 1, a(n) = 9*(10^(n-9) - a(n-9) + sum_{i=8..n-1} a(i)) for n>=9.

G.f.: x^8*(x-1)^2/((10*x-1)*(9*x^10-9*x^9+10*x-1)). - Alois P. Heinz, Feb 26 2015