A217776 a(n) = n*(n+1) + (n+2)*(n+3) + (n+4)*(n+5) + (n+6)*(n+7).

68, 100, 140, 188, 244, 308, 380, 460, 548, 644, 748, 860, 980, 1108, 1244, 1388, 1540, 1700, 1868, 2044, 2228, 2420, 2620, 2828, 3044, 3268, 3500, 3740, 3988, 4244, 4508, 4780, 5060, 5348, 5644, 5948, 6260, 6580, 6908, 7244, 7588, 7940, 8300, 8668, 9044, 9428

Offset: 0

Author

Jon Perry, Mar 24 2013

Keywords

• nonn
• easy

Examples

			a(1) = 1*2 + 3*4 + 5*6 + 7*8 = 2 + 12 + 30 + 56 = 100.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A020742, A027690, A051890 (two pairs), A217775 (3 pairs).

Programs

GAP

List([0..50], n-> (2*n+7)^2+19); # G. C. Greubel, Aug 27 2019

JavaScript

for (j=0;j<50;j++) {
a=j*(j+1)+(j+2)*(j+3)+(j+4)*(j+5)+(j+6)*(j+7);
document.write(a+", ");
}

Magma

[(2*n+7)^2+19: n in [0..50]]; // G. C. Greubel, Aug 27 2019

Maple

seq((2*n+7)^2+19, n=0..50); # G. C. Greubel, Aug 27 2019

Mathematica

(2*Range[50] +5)^2 +19 (* G. C. Greubel, Aug 27 2019 *)
Table[4n^2+28n+68,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{68,100,140},50] (* Harvey P. Dale, Jan 15 2020 *)

PARI

a(n)=4*n^2+28*n+68 \\ Charles R Greathouse IV, Jun 17 2017

Sage

[(2*n+7)^2+19 for n in (0..50)] # G. C. Greubel, Aug 27 2019

Formula

From Bruno Berselli, Mar 29 2013: (Start)

G.f.: 4*(17-26*x+11*x^2)/(1-x)^3.

a(n) = 4*n^2 + 28*n + 68.

a(n) = 4*A027690(n+3) = A020742(n)^2 + 19. (End)

E.g.f.: 4*(17 +8*x +x^2)*exp(x). - G. C. Greubel, Aug 27 2019