A380249 a(n) = 4^n - binomial(n,2)*3^(n-2).

1, 4, 15, 55, 202, 754, 2881, 11281, 45124, 183412, 753331, 3111739, 12879982, 53291398, 220074325, 906337909, 3721011016, 15228417832, 62133328423, 252794939071, 1025901734866, 4153971603034, 16786738847785, 67722274817305, 272813804258572

Offset: 0

Author

Enrique Navarrete, Feb 06 2025

Keywords

• nonn
• easy

Comments

a(n) is the number of words of length n defined on 4 letters where one of the letters is not used or is used any number of times except twice.

Examples

			For n=2, the 15 words on {0, 1, 2, 3} that do not use 0 exactly twice are 12, 21, 13, 31, 23, 32, 11, 22, 33, 10, 01, 20, 02, 30, 03.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (13,-63,135,-108).

Crossrefs

Cf. A086443, A380651, A380024.

Programs

Mathematica

LinearRecurrence[{13,-63,135,-108},{1,4,15,55},25] (* Stefano Spezia, Mar 03 2025 *)

Python

def A380249(n): return (1<<(n<<1))-(3**(n-2)*n*(n-1)>>1 if n>1 else 0) # Chai Wah Wu, Mar 15 2025

Formula

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

G.f.: (1 - 9*x + 26*x^2 - 23*x^3)/((1 - 3*x)^3*(1 - 4*x)). - Stefano Spezia, Mar 03 2025