A337313 a(n) is the number of n-digit positive integers with exactly three distinct base 10 digits.

0, 0, 648, 3888, 16200, 58320, 195048, 625968, 1960200, 6045840, 18468648, 56068848, 169533000, 511252560, 1539065448, 4627812528, 13904670600, 41756478480, 125354369448, 376232977008, 1129038669000, 3387795483600, 10164745404648, 30496954122288, 91496298184200

Offset: 1

Author

Stefano Spezia, Aug 22 2020

Keywords

• nonn
• easy
• base

Comments

a(n) is the number of n-digit numbers in A031962.

Examples

			a(1) = a(2) = 0 since the positive integers must have at least three digits;
a(3) = #{xyz in N | x,y,z are three different digits with x != 0} = 9*9*8 = 648;
a(4) = 3888 since #[9999] - #[999] - #(1111*[9]) - A335843(4) - #{xywz in N | x,y,w,z are four different digits with x != 0} = 9999 - 999 - 9 - 567 - 9*9*8*7 = 3888;
...
		

Links

• Index entries for linear recurrences with constant coefficients, signature (6,-11,6).
• Index entries for sequences related to digits.

Crossrefs

Cf. A000225, A000244, A000392, A031962, A048993, A335843, A337127.

Cf. A055642, A180599, A235155, A235691, A235718.

Programs

Mathematica

LinearRecurrence[{6,-11,6},{0,0,648},26]

PARI

concat([0,0],Vec(648*x^3/(1-6*x+11*x^2-6*x^3)+O(x^26)))

Formula

O.g.f.: 648*x^3/(1 - 6*x + 11*x^2 - 6*x^3).

E.g.f.: 108*(exp(x) - 1)^3.

a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) for n > 3.

a(n) = 648*S2(n, 3) where S2(n, 3) = A000392(n).

a(n) = 324*(3^(n-1) - 2^n + 1).

a(n) ~ 108 * 3^n.

a(n) = 324*(A000244(n-1) - A000225(n)).

a(n) = A337127(n, 3).