A056120 a(n) = (3^3)*4^(n-3) with a(0)=1, a(1)=1 and a(2)=7.

1, 1, 7, 27, 108, 432, 1728, 6912, 27648, 110592, 442368, 1769472, 7077888, 28311552, 113246208, 452984832, 1811939328, 7247757312, 28991029248, 115964116992, 463856467968, 1855425871872

Offset: 0

Author

Barry E. Williams, Jul 05 2000

Keywords

• easy
• nonn

Comments

For n>=3, a(n) is equal to the number of functions f:{1,2,...,n}->{1,2,3,4} such that for fixed, different x_1, x_2, x_3 in {1,2,...,n} and fixed y_1, y_2, y_3 in {1,2,3,4} we have f(x_i)<>y_i, (i=1,2,...,n). - Milan Janjic, May 13 2007

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
• Index entries for linear recurrences with constant coefficients, signature (4).

Crossrefs

Cf. A055841.

First differences of A002063.

Programs

GAP

Concatenation([1,1,7], List([3..25], n-> 27*4^(n-3) )); # G. C. Greubel, Jan 18 2020

Magma

[1,1,7] cat [27*4^(n-3): n in [3..25]]; // G. C. Greubel, Jan 18 2020

Maple

1,1,7, seq( 27*4^(n-3), n=3..25); # G. C. Greubel, Jan 18 2020

Mathematica

Table[If[n<2, 1, If[n==2, 7, 27*4^(n-3)]], {n,0,25}] (* G. C. Greubel, Jan 18 2020 *)

PARI

vector(26, n, if(n<2, 1, if(n==2, 7, 27*4^(n-3))) ) \\ G. C. Greubel, Jan 18 2020

Sage

[1,1,7]+[27*4^(n-3) for n in (3..25)] # G. C. Greubel, Jan 18 2020

Formula

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

G.f.: (1-x)^3/(1-4*x).

E.g.f.: (37 - 44*x + 8*x^2 + 27*exp(4*x))/64. - G. C. Greubel, Jan 18 2020

Extensions

a(21) corrected by R. J. Mathar, Dec 03 2014