A255375
Number of strings of n decimal digits that contain at least one string of exactly 5 consecutive "0" digits.
Original entry on oeis.org
0, 0, 0, 0, 0, 1, 18, 261, 3420, 42300, 504000, 5849991, 66599757, 746995383, 8279925399, 90898904070, 989984885400, 10709800740081, 115197460652916, 1232968482968526, 13139617033315116, 139495426762343121, 1475946175849599240, 15569374280153300271
Offset: 0
- Alois P. Heinz, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (20,-100,0,0,0,-9,99,-90).
Cf.
A255371-
A255374 (for the k=1 through k=4 cases of "exactly k consecutive '0' digits"),
A255376-
A255380 (for the cases k=6 through k=10).
-
LinearRecurrence[{20,-100,0,0,0,-9,99,-90},{0,0,0,0,0,1,18,261},30] (* Harvey P. Dale, Dec 12 2023 *)
A255377
Number of strings of n decimal digits that contain at least one string of exactly 7 consecutive "0" digits.
Original entry on oeis.org
0, 0, 0, 0, 0, 0, 0, 1, 18, 261, 3420, 42300, 504000, 5850000, 66600000, 746999991, 8279999757, 90899995383, 989999925399, 10709998904070, 115199984885400, 1232999800740000, 13139997460650000, 139499968482900081, 1475999617032002916, 15569995426740068526
Offset: 0
- Alois P. Heinz, 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,-9,99,-90).
Cf.
A255371-
A255376 (for the k=1 through k=6 cases of "exactly k consecutive '0' digits"),
A255378-
A255380 (for the cases k=8 through k=10).
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".
Showing 1-3 of 3 results.
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