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

0,2

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).

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)

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 *)

Prog

(PARI) a(n)=sum(i=0, n, 8*i^3); \\ Michael B. Porter, Mar 16 2015
(Magma) [2*n^2*(n+1)^2: n in [0..40]]; // Bruno Berselli, Mar 23 2015
(GAP) List([0..35], n->2*(n*(n+1))^2); # Muniru A Asiru, Oct 24 2018

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.

Sequence in context: A064015 A044576 A104453 * A143945 A239095 A189954

Adjacent sequences: A254368 A254369 A254370 * A254372 A254373 A254374

Keyword

nonn,easy

Author

Luciano Ancora, Mar 16 2015

Status

approved