This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A255381 #17 Jun 23 2015 05:47:16 %S A255381 1,18,261,3420,42300,504000,5850000,66600000,747000000,8280000000, %T A255381 90900000000,990000000000,10710000000000,115200000000000, %U A255381 1233000000000000,13140000000000000,139500000000000000,1476000000000000000,15570000000000000000,163800000000000000000 %N A255381 Number of strings of k+n decimal digits that contain one string of exactly k consecutive "0" digits, where k >= n. %C A255381 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.) %C A255381 Omitting the zero terms of each, A255371 through A255380 begin %C A255381 1, 18, 252, 3177, 37764, 432315, 4821867, 52767711, ... %C A255381 1, 18, 261, 3411, 42057, 499383, 5775480, 65506986, ... %C A255381 1, 18, 261, 3420, 42291, 503757, 5845383, 66525399, ... %C A255381 1, 18, 261, 3420, 42300, 503991, 5849757, 66595383, ... %C A255381 1, 18, 261, 3420, 42300, 504000, 5849991, 66599757, ... %C A255381 1, 18, 261, 3420, 42300, 504000, 5850000, 66599991, ... %C A255381 1, 18, 261, 3420, 42300, 504000, 5850000, 66600000, ... %C A255381 1, 18, 261, 3420, 42300, 504000, 5850000, 66600000, ... %C A255381 1, 18, 261, 3420, 42300, 504000, 5850000, 66600000, ... %C A255381 1, 18, 261, 3420, 42300, 504000, 5850000, 66600000, ... %C A255381 and the terms of this present sequence give the limiting value for each column. %H A255381 Colin Barker, <a href="/A255381/b255381.txt">Table of n, a(n) for n = 0..996</a> %H A255381 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (20,-100). %F A255381 a(0) = 1, a(n) = (81n + 99) * 10^(n-2) for n >= 1. %F A255381 G.f.: (x-1)^2/(10*x-1)^2. - _Alois P. Heinz_, Feb 27 2015 %F A255381 a(n) = 20*a(n-1) - 100*a(n-2) for n>2. - _Colin Barker_, Feb 27 2015 %e A255381 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. %e A255381 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". %e A255381 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". %e A255381 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". %o A255381 (PARI) Vec((x-1)^2/(10*x-1)^2 + O(x^100)) \\ _Colin Barker_, Feb 27 2015 %Y A255381 Cf. A255371, A255372, A255373, A255374, A255375, A255376, A255377, A255378, A255379, A255380. %K A255381 nonn,base,easy %O A255381 0,2 %A A255381 _Jon E. Schoenfield_, Feb 27 2015