A255380
Number of strings of n decimal digits that contain at least one string of exactly 10 consecutive "0" digits.
Original entry on oeis.org
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
- 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).
Cf.
A255371-
A255379 (for the k=1 through k=9 cases of "exactly k consecutive '0' digits").
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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 *)
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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
A255378
Number of strings of n decimal digits that contain at least one string of exactly 8 consecutive "0" digits.
Original entry on oeis.org
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
- Index entries for linear recurrences with constant coefficients, signature (20,-100,0,0,0,0,0,0,-9,99,-90).
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).
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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 *)
A255381
Number of strings of k+n decimal digits that contain one string of exactly k consecutive "0" digits, where k >= n.
Original entry on oeis.org
1, 18, 261, 3420, 42300, 504000, 5850000, 66600000, 747000000, 8280000000, 90900000000, 990000000000, 10710000000000, 115200000000000, 1233000000000000, 13140000000000000, 139500000000000000, 1476000000000000000, 15570000000000000000, 163800000000000000000
Offset: 0
Trivially, a(0)=1 because there is 1 string of k decimal digits that contains one string of exactly k consecutive "0" digits, where k >= 0: namely, the string of k consecutive "0" digits itself.
a(1)=18 because there are 18 strings of k+1 decimal digits that contain one string of exactly k consecutive "0" digits, where k >= 1. Letting "S" and "+" represent the string of exactly k consecutive "0" digits and any nonzero digit, respectively, the 18 strings comprise 9 of the form "S+" and 9 of the form "+S".
a(2)=261 because there are 261 strings of k+2 decimal digits that contain one string of exactly k consecutive "0" digits, where k >= 2. Letting "S", "+", and "." represent the string of exactly k consecutive "0" digits, any nonzero digit, and any digit (zero or nonzero), respectively, the 261 strings comprise 9*10=90 of the form "S+.", 9*9=81 of the form "+S+", and 10*9=90 of the form ".+S".
a(3)=3420 because there are 3420 strings of k+3 decimal digits that contain one string of exactly k consecutive "0" digits, where k >= 3. Using "S", "+", and "." as above, the 3420 strings comprise 9*10*10=900 of the form "S+..", 9*9*10=810 of the form "+S+.", 10*9*9=810 of the form ".+S+", and 10*10*9=900 of the form "..+S".
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