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

1,3

Comments

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

Links

Table of n, a(n) for n=1..25.

Index entries for linear recurrences with constant coefficients, signature (6,-11,6).

Index entries for sequences related to digits.

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).

Example

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;
...

Mathematica

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

Prog

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

Crossrefs

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

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

Sequence in context: A272780 A171606 A006914 * A204392 A233677 A200938

Adjacent sequences: A337310 A337311 A337312 * A337314 A337315 A337316

Keyword

nonn,easy,base

Author

Stefano Spezia, Aug 22 2020

Status

approved