A016765 Expansion of g.f. 1/((1-3*x)*(1-4*x)*(1-6*x)).

1, 13, 115, 865, 5971, 39193, 249355, 1555105, 9573091, 58428073, 354585595, 2143759345, 12928070611, 77832076153, 468051849835, 2812563019585, 16892428846531, 101422905135433, 608811146458075, 3653962903591825, 21928165007708851, 131586550851237913, 789589579708426315

Offset: 0

Author

N. J. A. Sloane, Dec 11 1999

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (13,-54,72).

Crossrefs

Cf. A016137, A016149.

Programs

Magma

[6^(n+1)-2^(2*n+3)+3^(n+1): n in [0..20]]; // Wesley Ivan Hurt, May 15 2014

Maple

A016765:=n->6^(n+1)-2^(2*n+3)+3^(n+1); seq(A016765(n), n=0..20); # Wesley Ivan Hurt, May 15 2014

Mathematica

Table[6^(n + 1) - 2^(2*n + 3) + 3^(n + 1), {n, 0, 20}] (* Wesley Ivan Hurt, May 15 2014 *)
CoefficientList[Series[1/((1-3x)(1-4x)(1-6x)),{x,0,30}],x] (* or *) LinearRecurrence[{13,-54,72},{1,13,115},30] (* Harvey P. Dale, Jul 18 2021 *)

PARI

vector(30,n,n--; 6^(n+1)-2^(2*n+3)+3^(n+1)) \\ G. C. Greubel, Sep 15 2018

Formula

From Vincenzo Librandi, Mar 20 2011: (Start)

a(n) = 6^(n+1) - 2^(2*n+3) + 3^(n+1).

a(n) = 10*a(n-1) - 24*a(n-2) + 3^n, n >= 2. (End)

G.f.: 1/((1-3*x)*(1-4*x)*(1-6*x)) = -3/(1-3*x) + 8/(1-4*x) - 6/(1-6*x). - Wolfdieter Lang, May 19 2014

From Elmo R. Oliveira, Mar 26 2025: (Start)

E.g.f.: exp(3*x)*(6*exp(3*x) - 8*exp(x) + 3).

a(n) = 13*a(n-1) - 54*a(n-2) + 72*a(n-3).

a(n) = A016149(n+1) - A016137(n+1). (End)