A085788 Partial sums of n 3-spaced triangular numbers beginning with t(3), e.g., a(2) = t(3)+t(6) = 6+21 = 27.

6, 27, 72, 150, 270, 441, 672, 972, 1350, 1815, 2376, 3042, 3822, 4725, 5760, 6936, 8262, 9747, 11400, 13230, 15246, 17457, 19872, 22500, 25350, 28431, 31752, 35322, 39150, 43245, 47616, 52272, 57222, 62475, 68040, 73926, 80142, 86697, 93600, 100860, 108486

Offset: 1

Author

Jon Perry, Jul 23 2003

Keywords

• nonn
• easy

Links

• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A004188, A006002.

Row sums of triangle A001283.

Cf. A254407. - Bruno Berselli, Jan 30 2015

Programs

Maple

a:=n->sum(sum(sum(j-k+1, j=1..n), k=0..n),m=0..n): seq(a(n), n=1..45); # Zerinvary Lajos, May 30 2007

Mathematica

LinearRecurrence[{4,-6,4,-1},{6,27,72,150},50] (* Harvey P. Dale, Dec 14 2017 *)

PARI

v=vector(40,i,i*(i+1)/2); s=0; forstep(i=3,40,3,s+=v[i]; print1(s","))

Formula

a(n) = (3/2)*n*(n+1)^2 = 3*A006002(n).

a(n) = Sum_{j=1..n} (j+n+1)*(n+1). - Zerinvary Lajos, Sep 10 2006

From Colin Barker, Mar 17 2014: (Start)

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).

G.f.: 3*x*(x+2)/(x-1)^4. (End)

E.g.f.: 3*exp(x)*x*(1 + x)*(4 + x)/2. - Elmo R. Oliveira, Aug 14 2025

Extensions

Edited and more terms from Michel Marcus, Mar 17 2014