A365137 - OEIS

%I #25 Aug 24 2023 10:14:09

%S 0,0,1,18,261,3411,42048,499131,5770611,65427678,730784601,8065910511,

%T 88170256008,956125498671,10298661792111,110293085617038,

%U 1175325726682341,12470569310694411,131813055336390768,1388552621823766611,14583291094441416411,152746593446386647198

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

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

%H Felix Huber, <a href="/A365137/b365137.txt">Table of n, a(n) for n = 0..996</a>

%H Armin Widmer, <a href="/A365137/a365137_1.pdf">The number of Licence Plates that contain '22'</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (19,-81,-90).

%F 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.

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

%F 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.

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

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

%e a(2) = 1, the number 22 itself.

%e a(3) = 18, 10 numbers 22X plus 9 numbers X22 minus 1 number 222.

%e a(4) = 261, 100 numbers 22XX plus 90 numbers X22X plus 90 numbers XX22 minus 10 numbers 222X minus 9 numbers X222.

%p 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);

%t LinearRecurrence[{19, -81, -90}, {0, 0, 1, 18}, 22] (* _Robert P. P. McKone_, Aug 24 2023 *)

%Y Cf. A057092, A255372.

%K nonn,base,easy

%O 0,4

%A _Felix Huber_, Aug 23 2023