A135037 Sums of the products of n consecutive triples of numbers.

0, 60, 396, 1386, 3570, 7650, 14490, 25116, 40716, 62640, 92400, 131670, 182286, 246246, 325710, 423000, 540600, 681156, 847476, 1042530, 1269450, 1531530, 1832226, 2175156, 2564100, 3003000, 3495960, 4047246, 4661286, 5342670

Offset: 1

Author

Cino Hilliard, Feb 10 2008

Keywords

• nonn
• easy

Examples

			For n = 3, the sum of the first 3 triples is 0*1*2+3*4*5+6*7*8 =396, the 3rd entry in the sequence.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Programs

Magma

[(27*n^4-18*n^3-15*n^2+6*n)/4: n in [1..40]]; // Vincenzo Librandi, Sep 18 2016

Mathematica

Table[(27 n^4 - 18 n^3 - 15 n^2 + 6 n)/4, {n, 1, 50}] (* or *) LinearRecurrence[{5,-10,10,-5,1}, {0, 60, 396, 1386, 3570}, 25] (* G. C. Greubel, Sep 17 2016 *)

PARI

sumprod3(n) = { local(x,s=0); forstep(x=0,n,3, s+=x*(x+1)*(x+2); print1(s",") ) }

Formula

a(1) = 0*1*2, a(2) = 0*1*2 + 3*4*5, ..., a(n) = 0*1*2 + 3*4*5 + 6*7*8 + ... + (2n-1)*(2n)*(2n+1).

a(n) = (27*n^4 - 18*n^3 - 15*n^2 + 6*n)/4.

From R. J. Mathar, Feb 14 2008: (Start)

O.g.f.: 6*x^2*(10+16*x+x^2)/(1-x)^5.

a(n) = 6*A024391(n-1). (End)

E.g.f.: (3/4)*x^2*(40 + 48*x + 9*x^2)*exp(x). - G. C. Greubel, Sep 17 2016