A337314 a(n) is the number of n-digit positive integers with exactly four distinct base 10 digits.

0, 0, 0, 4536, 45360, 294840, 1587600, 7715736, 35244720, 154700280, 661122000, 2773768536, 11487556080, 47136955320, 192126589200, 779279814936, 3149513947440, 12695388483960, 51073849285200, 205172877726936, 823325141746800, 3301203837670200, 13228529919066000

Offset: 1

Author

Stefano Spezia, Sep 26 2020

Keywords

• nonn
• easy
• base

Comments

a(n) is the number of n-digit numbers in A031969.

Examples

			a(1) = a(2) = a(3) = 0 since the positive integers must have at least four digits;
a(4) = #{wxyz in N | w,x,y,z are four different digits with w != 0} = A073531(4) = 4536;
a(5) = 45360 since #[99999] - #[9999] - #(11111*[9]) - A335843(5) - A337313(5) - #{vwxyz in N | v,w,x,y,z are five different digits with v != 0} = 99999 - 9999 - 9 - 1215 - 16200 - 9*9*8*7*6 = 45360;
...
		

Links

• Index entries for linear recurrences with constant coefficients, signature (10,-35,50,-24).
• Index entries for sequences related to digits.

Crossrefs

Cf. A000079, A000244, A000302, A000453, A031969, A073531, A335843, A337313, A337127.

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

Programs

Mathematica

LinearRecurrence[{10,-35,50,-24},{0,0,0,4536},23]

PARI

concat([0,0,0],Vec(4536*x^4/(1-10*x+35*x^2-50*x^3+24*x^4)+O(x^24)))

Formula

O.g.f.: 4536*x^4/(1 - 10*x + 35*x^2 - 50*x^3 + 24*x^4).

E.g.f.: 189*(exp(x) - 1)^4.

a(n) = 10*a(n-1) - 35*a(n-2) + 50*a(n-3) - 24*a(n-4) for n > 4.

a(n) = 4536*S2(n, 4) where S2(n, 4) = A000453(n).

a(n) = 189*(4^n - 4*3^n + 3*2^(n+1) - 4).

a(n) ~ 189 * 4^n.

a(n) = 189*(A000302(n) - 4*A000244(n) + 3*A000079(n+1) - 4).

a(n) = A337127(n, 4).