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A255380
Number of strings of n decimal digits that contain at least one string of exactly 10 consecutive "0" digits.
11
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
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
KEYWORD
nonn,base,easy
AUTHOR
Jon E. Schoenfield, Feb 22 2015
STATUS
approved