A335843 a(n) is the number of n-digit positive integers with exactly two distinct base 10 digits.
0, 81, 243, 567, 1215, 2511, 5103, 10287, 20655, 41391, 82863, 165807, 331695, 663471, 1327023, 2654127, 5308335, 10616751, 21233583, 42467247, 84934575, 169869231, 339738543, 679477167, 1358954415, 2717908911, 5435817903, 10871635887, 21743271855, 43486543791
Offset: 1
Examples
a(1) = 0 since the positive integers must have at least two digits;
a(2) = 81 since #[99] - #[9] - #(11*[9]) = 99 - 9 - 9 = 81;
a(3) = 243 since #[999] - #[99] - #(111*[9]) - #{xyz in N | x,y,z are three different digits with x != 0} = 999 - 99 - 9 - 9*9*8 = 243;
...
Links
- Stefano Spezia, Table of n, a(n) for n = 1..3300
- Puzzle Critic, Twitter post about the case n = 5, Jul 16 2020.
- Index entries for linear recurrences with constant coefficients, signature (3,-2).
- Index entries for sequences related to digits.
Programs
-
Mathematica
LinearRecurrence[{3,-2},{0,81},31] -
PARI
concat([0],Vec(81*x^2/(1-3*x+2*x^2)+O(x^31)))
Formula
O.g.f.: 81*x^2/(1 - 3*x + 2*x^2).
E.g.f.: 81*(exp(x) - 1)^2/2.
a(n) = 3*a(n-1) - 2*a(n-2) for n > 2.
a(n) = 81*(2^(n-1) - 1).
a(n) = 81*A000225(n-1).
Extensions
a(0) removed by Stefano Spezia, Sep 23 2020
Comments