A254371 Sum of cubes of the first n even numbers (A016743).

0, 8, 72, 288, 800, 1800, 3528, 6272, 10368, 16200, 24200, 34848, 48672, 66248, 88200, 115200, 147968, 187272, 233928, 288800, 352800, 426888, 512072, 609408, 720000, 845000, 985608, 1143072, 1318688, 1513800, 1729800, 1968128, 2230272, 2517768, 2832200, 3175200

Offset: 0

Author

Luciano Ancora, Mar 16 2015

Keywords

• nonn
• easy

Comments

Property: for n >= 2, each (a(n), a(n)+1, a(n)+2) is a triple of consecutive terms that are the sum of two nonzero squares; precisely: a(n) = (n*(n + 1))^2 + (n*(n + 1))^2, a(n)+1 = (n^2+2n)^2 + (n^2-1)^2 and a(n)+2 = (n^2+n+1)^2 + (n^2+n-1)^2 (see Diophante link). - Bernard Schott, Oct 05 2021

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Luciano Ancora, The Square Pyramidal Number and other figurate numbers, ch. 3.
• Diophante, Une miniature avec trois entiers consécutifs (in French).
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A000537 (sum of first n cubes); A002593 (sum of first n odd cubes).

Cf. A060300 (2*a(n)).

First bisection of A105636; second bisection of A212892.

Cf. A000217, A000415, A002378, A035287, A163102.

Programs

GAP

List([0..35],n->2*(n*(n+1))^2); # Muniru A Asiru, Oct 24 2018

Magma

[2*n^2*(n+1)^2: n in [0..40]]; // Bruno Berselli, Mar 23 2015

Maple

A254371:=n->2*n^2*(n + 1)^2: seq(A254371(n), n=0..50); # Wesley Ivan Hurt, Apr 28 2017

Mathematica

Table[2 n^2 (n+1)^2, {n, 0, 40}] (* or *) LinearRecurrence[{5, -10, 10, -5, 1}, {0, 8, 72, 288, 800}, 40]
Accumulate[Range[0,80,2]^3] (* Harvey P. Dale, Jun 26 2017 *)

PARI

a(n)=sum(i=0, n, 8*i^3); \\ Michael B. Porter, Mar 16 2015

Formula

G.f.: 8*x*(1 + 4*x + x^2)/(1 - x)^5.

a(n) = 2*n^2*(n + 1)^2.

a(n) = 2*A035287(n+1) = 2*A002378(n)^2 = 8*A000217(n)^2. - Bruce J. Nicholson, Apr 23 2017

a(n) = 8*A000537(n). - Michel Marcus, Apr 23 2017

From Amiram Eldar, Aug 25 2022: (Start)

Sum_{n>=1} 1/a(n) = Pi^2/6 - 3/2.

Sum_{n>=1} (-1)^(n+1)/a(n) = 3/2 - 2*log(2). (End)

From Elmo R. Oliveira, Aug 14 2025: (Start)

E.g.f.: 2*x*(2 + x)*(2 + 6*x + x^2)*exp(x).

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).

a(n) = 4*A163102(n) = A060300(n)/2. (End)