A365137 a(n) is the number of n-digit numbers that contain '22' in their decimal representation.

0, 0, 1, 18, 261, 3411, 42048, 499131, 5770611, 65427678, 730784601, 8065910511, 88170256008, 956125498671, 10298661792111, 110293085617038, 1175325726682341, 12470569310694411, 131813055336390768, 1388552621823766611, 14583291094441416411, 152746593446386647198

Offset

0,4

Comments

a(n) is also valid for '11', '33', '44', '55', '66', '77', '88' or '99' instead of '22'.

Links

Felix Huber, Table of n, a(n) for n = 0..996

Armin Widmer, The number of Licence Plates that contain '22'

Index entries for linear recurrences with constant coefficients, signature (19,-81,-90).

Formula

a(n) = 19*a(n - 1) - 81*a(n - 2) - 90*a(n - 3) with a(0) = a(1) = 0, a(2) = 1 and a(3) = 18 for n >= 4.

a(n) = 9*10^(n - 1) - A057092(n) + A057092(n - 2) with a(0) = a(1) = 0 for n >= 2.

a(0) = 0, a(n) = 9*10^(n - 1) - (p^(n + 1) - q^(n + 1))/(3*sqrt(13)) + (p^(n - 1) - q^(n - 1))/(3*sqrt(13)) with p = (9 + 3*sqrt(13))/2 and q = (9 - 3*sqrt(13))/2 for n >= 1.

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

a(n) = A255372(n) for n <= 5.

Example

a(2) = 1, the number 22 itself.
a(3) = 18, 10 numbers 22X plus 9 numbers X22 minus 1 number 222.
a(4) = 261, 100 numbers 22XX plus 90 numbers X22X plus 90 numbers XX22 minus 10 numbers 222X minus 9 numbers X222.

Maple

A365137 := proc(n) option remember; if n <= 1 then 0; elif n = 2 then 1; elif n = 3 then 18; else 19*procname(n - 1) - 81*procname(n - 2) - 90*procname(n - 3); end if; end proc; seq(A365137(n), n = 0 .. 21);

Mathematica

LinearRecurrence[{19, -81, -90}, {0, 0, 1, 18}, 22] (* Robert P. P. McKone, Aug 24 2023 *)

Crossrefs

Cf. A057092, A255372.

Sequence in context: A374880 A374862 A362808 * A255372 A255373 A255374

Adjacent sequences: A365134 A365135 A365136 * A365138 A365139 A365140

Keyword

nonn,base,easy

Author

Felix Huber, Aug 23 2023

Status

approved