A381646 a(n) = 4^n - 2*3^(n-1)*(n+3) + 2^(n-2)*(n^2+3*n+4).

0, 0, 0, 0, 6, 80, 650, 4172, 23310, 119016, 571122, 2621828, 11651222, 50536928, 215219706, 903799548, 3754755102, 15469272536, 63320624642, 257886717812, 1046169235110, 4230947198160, 17069749295370, 68738191563500, 276393979740206

Offset: 0

Author

Enrique Navarrete, Mar 03 2025

Keywords

• nonn
• easy

Comments

a(n) is the number of words of length n defined on 4 letters where two of the letters are used at least twice.

Examples

			For n=5 the 80 words that use 0 and 1 at least twice are 00111 (10 of this type), 00011 (10 of this type), 00112 (30 of this type), 00113 (30 of this type).
		

Links

• Index entries for linear recurrences with constant coefficients, signature (16,-105,362,-692,696,-288).

Crossrefs

Cf. A112495, A380024, A380249.

Programs

Mathematica

LinearRecurrence[{16,-105,362,-692,696,-288},{0,0,0,0,6,80},25] (* Stefano Spezia, Mar 03 2025 *)

Python

def A381646(n): return ((1<2 else 0 # Chai Wah Wu, Mar 15 2025

Formula

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

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

G.f.: 2*x^4*(3 - 8*x)/((1 - 3*x)^2*(1 - 2*x)^3*(1 - 4*x)). - Stefano Spezia, Mar 03 2025