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

1,4

Comments

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

Links

Table of n, a(n) for n=1..23.

Index entries for linear recurrences with constant coefficients, signature (10,-35,50,-24).

Index entries for sequences related to digits.

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).

Example

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;
...

Mathematica

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

Prog

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

Crossrefs

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

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

Sequence in context: A138731 A250450 A339795 * A234171 A262848 A307919

Adjacent sequences: A337311 A337312 A337313 * A337315 A337316 A337317

Keyword

nonn,easy,base

Author

Stefano Spezia, Sep 26 2020

Status

approved