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

1,2

Comments

a(n) is the number of n-digit numbers in A031955.

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.

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).

Example

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;
...

Mathematica

LinearRecurrence[{3, -2}, {0, 81}, 31]

Prog

(PARI) concat([0], Vec(81*x^2/(1-3*x+2*x^2)+O(x^31)))

Crossrefs

Cf. A000225, A031955, A337127, A337313.

Cf. A055642, A235154, A235690, A235717.

Sequence in context: A184068 A211690 A205055 * A255625 A205048 A008876

Adjacent sequences: A335840 A335841 A335842 * A335844 A335845 A335846

Keyword

nonn,easy,base

Author

Stefano Spezia, Jul 18 2020

Extensions

a(0) removed by Stefano Spezia, Sep 23 2020

Status

approved