A255381 - OEIS

A255381

Number of strings of k+n decimal digits that contain one string of exactly k consecutive "0" digits, where k >= n.

1, 18, 261, 3420, 42300, 504000, 5850000, 66600000, 747000000, 8280000000, 90900000000, 990000000000, 10710000000000, 115200000000000, 1233000000000000, 13140000000000000, 139500000000000000, 1476000000000000000, 15570000000000000000, 163800000000000000000 (list; graph; refs; listen; history; text; internal format)

	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.) `Omitting the zero terms of each, A255371 through A255380 begin` `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/(10x-1)^2. - Alois P. Heinz, Feb 27 2015` `a(n) = 20a(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 910=90 of the form "S+.", 99=81 of the form "+S+", and 109=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 91010=900 of the form "S+..", 9910=810 of the form "+S+.", 1099=810 of the form ".+S+", and 1010*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	`Cf. A255371, A255372, A255373, A255374, A255375, A255376, A255377, A255378, A255379, A255380.` `Sequence in context: A255378 A255379 A255380 * A078205 A316698 A254248` `Adjacent sequences: A255378 A255379 A255380 * A255382 A255383 A255384`
	KEYWORD	`nonn,base,easy`
	AUTHOR	`Jon E. Schoenfield, Feb 27 2015`
	STATUS	`approved`