A322053 -id:A322053 - cp's OEIS frontend

A322052 Number of decimal strings of length n that contain a specific string xy where x and y are distinct digits.

0, 1, 20, 299, 3970, 49401, 590040, 6850999, 77919950, 872348501, 9645565060, 105583302099, 1146187455930, 12356291257201, 132416725116080, 1411810959903599, 14985692873919910, 158445117779295501, 1669465484919035100, 17536209731411055499, 183692631829191519890, 1919390108560504143401

Offset: 1

N. J. A. Sloane, Dec 21 2018

See A004189 for the number that do not contain the specified string.

			Suppose the desired string is 03. At length 2 that is the only possibility. At length 3 there are 20 strings that contain it: 03d and d03, where d is any digit.

Robert Israel, Table of n, a(n) for n = 1..999
Index entries for linear recurrences with constant coefficients, signature (20,-101,10).

Cf. A004189, A322053, A322054.

Partial sums of A322628.

f:= gfun:-rectoproc({10*a(n) - 101*a(n + 1) + 20*a(n + 2) - a(n + 3), a(0) = 0, a(1) = 0, a(2) = 1},a(n),remember):
map(f, [$1..40]); # Robert Israel, Mar 27 2020

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

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

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Dec 21 2018

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

			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.

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).

Cf. A322052, A004189, A322053.

Suggested by A322628.

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

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.

cp's OEIS Frontend

A322052 Number of decimal strings of length n that contain a specific string xy where x and y are distinct digits.

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