A385329 a(n) = 5^n - 2*4^(n-1)*(n+4) + 3^(n-2)*(n^2+5*n+9).

0, 0, 0, 0, 6, 110, 1220, 10612, 79786, 544434, 3468792, 21012200, 122500334, 693324502, 3833742796, 20809676604, 111288341970, 588046458074, 3076991784512, 15972440574064, 82370489136214, 422506631928510, 2157589903432020, 10977781519321220, 55686118748465786

Offset

0,5

Comments

a(n) is the number of words of length n defined on 5 letters where two chosen letters, say a and b, are used at least twice.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..500

Index entries for linear recurrences with constant coefficients, signature (22,-200,962,-2583,3672,-2160).

Formula

E.g.f.: exp(3*x)*(exp(x) - x - 1)^2.

G.f.: 2*x^4*(3 - 11*x)/((1 - 4*x)^2*(1 - 3*x)^3*(1 - 5*x)). - Jinyuan Wang, Jun 26 2025

Example

a(4) = 6 since the words are the 6 permutations of aabb.
a(5) = 110 since the words are (number of permutations in parentheses): aaabb (10), aabbb (10), aabbc (30), aabbd (30), aabbe (30).

Mathematica

LinearRecurrence[{22, -200, 962, -2583, 3672, -2160}, {0, 0, 0, 0, 6, 110, 1220}, 25] (* Amiram Eldar, Jun 28 2025 *)

Prog

(Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); [0, 0, 0, 0] cat Coefficients(R!((2*x^4*(3 - 11*x)/((1 - 4*x)^2*(1 - 3*x)^3*(1 - 5*x))))); // Vincenzo Librandi, Jul 05 2025

Crossrefs

Cf. A112495, A381646.

Sequence in context: A217987 A119814 A227443 * A050884 A156554 A197325

Adjacent sequences: A385326 A385327 A385328 * A385330 A385331 A385332

Keyword

nonn,easy

Author

Enrique Navarrete, Jun 25 2025

Status

approved