A178977 a(n) = (3*n+2)*(3*n+5)/2.

5, 20, 44, 77, 119, 170, 230, 299, 377, 464, 560, 665, 779, 902, 1034, 1175, 1325, 1484, 1652, 1829, 2015, 2210, 2414, 2627, 2849, 3080, 3320, 3569, 3827, 4094, 4370, 4655, 4949, 5252, 5564, 5885, 6215, 6554, 6902, 7259, 7625, 8000, 8384, 8777, 9179, 9590, 10010

Offset: 0

Author

Paul Curtz, Jan 02 2011

Keywords

• nonn
• easy

Comments

Companion to A145910.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A008585, A145910, A168233, A168300, A016789, A178971, A235332.

Programs

Magma

[n*(n+3)/2: n in [2..135 by 3]]; // Bruno Berselli, Apr 14 2011

Maple

A178977:=n->(3*n+2)*(3*n+5)/2: seq(A178977(n), n=0..50); # Wesley Ivan Hurt, Oct 23 2014

Mathematica

f[n_] := (3 n + 2) (3 n + 5)/2; Array[f, 45, 0]
LinearRecurrence[{3,-3,1},{5,20,44},50] (* Harvey P. Dale, Apr 19 2013 *)

PARI

a(n)=(3*n+2)*(3*n+5)/2 \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n) = a(n-1) + 6 + 9*n.

a(n) = A178971(3*n+2).

a(n) = A145910(n) + 3 + 3*n = A145910(n) + A008585(n+1).

a(n) = A168233(n+1)*A168300(n+1).

G.f.: (-5-5*x+x^2)/(x-1)^3. [Adapted to the offset by Bruno Berselli, Apr 14 2011]

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Apr 19 2013

From Amiram Eldar, Mar 10 2022: (Start)

Sum_{n>=0} 1/a(n) = 1/3.

Sum_{n>=0} (-1)^n/a(n) = 4*Pi/(9*sqrt(3)) - 1/3 - 4*log(2)/9. (End)

From Elmo R. Oliveira, Oct 30 2024: (Start)

E.g.f.: exp(x)*exp(x)*(5 + 15*x + 9*x^2/2).

a(n) = A016789(n)*A016789(n+1)/2. (End)