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) = 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
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. A060300 (2*a(n)).
KEYWORD
nonn,easy
AUTHOR
Luciano Ancora, Mar 16 2015
STATUS
approved