A062026 a(n) = n*(n+1)*(n^2 - 3*n + 6)/4.

0, 2, 6, 18, 50, 120, 252, 476, 828, 1350, 2090, 3102, 4446, 6188, 8400, 11160, 14552, 18666, 23598, 29450, 36330, 44352, 53636, 64308, 76500, 90350, 106002, 123606, 143318, 165300, 189720, 216752, 246576, 279378, 315350, 354690, 397602, 444296

Offset: 0

Author

Amarnath Murthy, Jun 02 2001

Keywords

• nonn
• easy

Comments

a(n) = 1*2*3 + 2*3*4 + 3*4*5 +. . .+ (n-2)*(n-1)*n +(n-1)*n*1+ n*1*2, the sum of the cyclic product of terms taken three at a time, final term being n*1*2=2n.

Examples

			a(4) = 1*2*3 + 2*3*4 + 3*4*1 + 4*1*2 = 50.
		

Links

• Harry J. Smith, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A004255.

Programs

Mathematica

Table[n(n+1)(n^2-3n+6)/4,{n,0,40}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{0,2,6,18,50},40] (* Harvey P. Dale, Apr 22 2015 *)

PARI

a(n) = n*(n+1)*(n^2 -3*n +6)/4 \\ Harry J. Smith, Jul 29 2009

SageMath

[n*(n+1)*(n^2-3*n+6)/4 for n in (0..40)] # G. C. Greubel, May 05 2022

Formula

a(n) = 2 * A004255(n).

a(0)=0, a(1)=2, a(2)=6, a(3)=18, a(4)=50, a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Harvey P. Dale, Apr 22 2015

From G. C. Greubel, May 05 2022: (Start)

a(n) = 6*binomial(n+3, 4) - 12*binomial(n+2, 3) + 8*binomial(n+1, 2).

G.f.: 2*x*(1 - 2*x + 4*x^2)/(1-x)^5.

E.g.f.: (1/4)*x*(8 + 4*x + 4*x^2 + x^3)*exp(x). (End)

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Jun 06 2001