A322054 Number of decimal strings of length n that do not contain a specific string xx (where x is a single digit).

10, 99, 981, 9720, 96309, 954261, 9455130, 93684519, 928256841, 9197472240, 91131561729, 902961305721, 8946835807050, 88648174014939, 878355088397901, 8703029361715560, 86232460051021149, 854419404714630381, 8465866782890863770

Offset: 1

Author

N. J. A. Sloane, Dec 21 2018

Keywords

• nonn
• base
• easy

Comments

See A322053 for the number that do contain the specified string.

Examples

			Suppose the string is 00. At length 2 there are 99 strings that do not contain it. At length 3 there are 19 strings that do not contain it, 000, 00x, and x00, where x is any nonzero digit. So a(3) = 1000-19 = 981.
		

Links

• Robert P. P. McKone, Table of n, a(n) for n = 1..1000
• Jean-Paul Allouche, Jeffrey Shallit, and Manon Stipulanti, Combinatorics on words and generating Dirichlet series of automatic sequences, arXiv:2401.13524 [math.CO], 2024.
• Index entries for linear recurrences with constant coefficients, signature (9,9).

Crossrefs

Cf. A322052, A004189, A322053.

Suggested by A322628.

Programs

Mathematica

T[n_, k_] := LinearRecurrence[{n - 1, n - 1}, {n, n^2 - 1}, k];
T[10, {1, 19}] (* Robert P. P. McKone, Dec 31 2020 *)

Formula

G.f.: x*(10+9*x)/(1-9*x-9*x^2).

a(n) = 9*a(n-1) + 9*a(n-2) for n >= 3.