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

%I A254371 #76 Aug 15 2025 09:12:13
%S A254371 0,8,72,288,800,1800,3528,6272,10368,16200,24200,34848,48672,66248,
%T A254371 88200,115200,147968,187272,233928,288800,352800,426888,512072,609408,
%U A254371 720000,845000,985608,1143072,1318688,1513800,1729800,1968128,2230272,2517768,2832200,3175200
%N A254371 Sum of cubes of the first n even numbers (A016743).
%C A254371 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
%H A254371 Vincenzo Librandi, <a href="/A254371/b254371.txt">Table of n, a(n) for n = 0..1000</a>
%H A254371 Luciano Ancora, <a href="https://upload.wikimedia.org/wikipedia/commons/9/9c/FigurateN.pdf">The Square Pyramidal Number and other figurate numbers</a>, ch. 3.
%H A254371 Diophante, <a href="http://www.diophante.fr/problemes-par-themes/arithmetique-et-algebre/a1-pot-pourri/983-a1931-une-miniature-avec-trois-entiers-consecutifs">Une miniature avec trois entiers consécutifs</a> (in French).
%H A254371 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A254371 G.f.: 8*x*(1 + 4*x + x^2)/(1 - x)^5.
%F A254371 a(n) = 2*n^2*(n + 1)^2.
%F A254371 a(n) = 2*A035287(n+1) = 2*A002378(n)^2 = 8*A000217(n)^2. - _Bruce J. Nicholson_, Apr 23 2017
%F A254371 a(n) = 8*A000537(n). - _Michel Marcus_, Apr 23 2017
%F A254371 From _Amiram Eldar_, Aug 25 2022: (Start)
%F A254371 Sum_{n>=1} 1/a(n) = Pi^2/6 - 3/2.
%F A254371 Sum_{n>=1} (-1)^(n+1)/a(n) = 3/2 - 2*log(2). (End)
%F A254371 From _Elmo R. Oliveira_, Aug 14 2025: (Start)
%F A254371 E.g.f.: 2*x*(2 + x)*(2 + 6*x + x^2)*exp(x).
%F A254371 a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
%F A254371 a(n) = 4*A163102(n) = A060300(n)/2. (End)
%p A254371 A254371:=n->2*n^2*(n + 1)^2: seq(A254371(n), n=0..50); # _Wesley Ivan Hurt_, Apr 28 2017
%t A254371 Table[2 n^2 (n+1)^2, {n, 0, 40}] (* or *) LinearRecurrence[{5, -10, 10, -5, 1}, {0, 8, 72, 288, 800}, 40]
%t A254371 Accumulate[Range[0,80,2]^3] (* _Harvey P. Dale_, Jun 26 2017 *)
%o A254371 (PARI) a(n)=sum(i=0, n, 8*i^3); \\ _Michael B. Porter_, Mar 16 2015
%o A254371 (Magma) [2*n^2*(n+1)^2: n in [0..40]]; // _Bruno Berselli_, Mar 23 2015
%o A254371 (GAP) List([0..35],n->2*(n*(n+1))^2); # _Muniru A Asiru_, Oct 24 2018
%Y A254371 Cf. A000537 (sum of first n cubes); A002593 (sum of first n odd cubes).
%Y A254371 Cf. A060300 (2*a(n)).
%Y A254371 First bisection of A105636; second bisection of A212892.
%Y A254371 Cf. A000217, A000415, A002378, A035287, A163102.
%K A254371 nonn,easy
%O A254371 0,2
%A A254371 _Luciano Ancora_, Mar 16 2015