OFFSET
0,2
COMMENTS
This sequence gives the first k+1 nonzero terms from A255371 (the k=1 case), A255372 (the k=2 case), etc., through A255380 (the k=10 case). Those sequences' definitions concern the number of strings of decimal digits that contain "at least one" string of exactly k consecutive "0" digits; the present sequence omits the words "at least" because, with k >= n, and thus 2k >= k+n, it is not possible to have more than one string of exactly k consecutive "0" digits in a string of k+n digits. (Two strings each having exactly k consecutive "0" digits would have to be separated by at least one nonzero digit, or else they would constitute a single string of exactly 2k consecutive "0" digits.)
1, 18, 252, 3177, 37764, 432315, 4821867, 52767711, ...
1, 18, 261, 3411, 42057, 499383, 5775480, 65506986, ...
1, 18, 261, 3420, 42291, 503757, 5845383, 66525399, ...
1, 18, 261, 3420, 42300, 503991, 5849757, 66595383, ...
1, 18, 261, 3420, 42300, 504000, 5849991, 66599757, ...
1, 18, 261, 3420, 42300, 504000, 5850000, 66599991, ...
1, 18, 261, 3420, 42300, 504000, 5850000, 66600000, ...
1, 18, 261, 3420, 42300, 504000, 5850000, 66600000, ...
1, 18, 261, 3420, 42300, 504000, 5850000, 66600000, ...
1, 18, 261, 3420, 42300, 504000, 5850000, 66600000, ...
and the terms of this present sequence give the limiting value for each column.
LINKS
Colin Barker, Table of n, a(n) for n = 0..996
Index entries for linear recurrences with constant coefficients, signature (20,-100).
FORMULA
a(0) = 1, a(n) = (81n + 99) * 10^(n-2) for n >= 1.
G.f.: (x-1)^2/(10*x-1)^2. - Alois P. Heinz, Feb 27 2015
a(n) = 20*a(n-1) - 100*a(n-2) for n>2. - Colin Barker, Feb 27 2015
EXAMPLE
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".
PROG
(PARI) Vec((x-1)^2/(10*x-1)^2 + O(x^100)) \\ Colin Barker, Feb 27 2015
CROSSREFS
KEYWORD
nonn,base,easy
AUTHOR
Jon E. Schoenfield, Feb 27 2015
STATUS
approved