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%I #65 Jan 10 2023 07:59:43
%S 0,2,16,54,128,250,432,686,1024,1458,2000,2662,3456,4394,5488,6750,
%T 8192,9826,11664,13718,16000,18522,21296,24334,27648,31250,35152,
%U 39366,43904,48778,54000,59582,65536,71874,78608,85750,93312,101306,109744,118638,128000,137842
%N a(n) = 2*n^3.
%C Also the largest determinant of a 3 X 3 matrix with entries from {0..n}. - _Jud McCranie_, Aug 12 2001
%C 4*a(n) is a perfect cube.
%C The positive terms comprise the principal diagonal of the convolution array A213821. - _Clark Kimberling_, Jul 04 2012
%C Volume of a pyramid (square base) with side n and height 6*n. - _Wesley Ivan Hurt_, Aug 25 2014
%H Vincenzo Librandi, <a href="/A033431/b033431.txt">Table of n, a(n) for n = 0..10000</a>
%H Amelia Carolina Sparavigna, <a href="https://doi.org/10.5281/zenodo.4662348">Generalized Sum of Stella Octangula Numbers</a>, Politecnico di Torino (Italy, 2021).
%H Amelia Carolina Sparavigna, <a href="https://doi.org/10.5281/zenodo.4663050">Cardano Formula and Some Figurate Numbers</a>, Politecnico di Torino (Italy, 2021).
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F G.f.: 2*x*(1 + 4*x + x^2) / (1 - x)^4. - _R. J. Mathar_, Feb 04 2011
%F a(n) = 2*A000578(n). - _Omar E. Pol_, May 14 2008
%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - _Wesley Ivan Hurt_, Aug 25 2014
%F a(n) = A002378(n)^2 - A002378(n^2). - _Bruno Berselli_, Oct 20 2016
%F E.g.f.: 2*x*(1 + 3*x + x^2)*exp(x). - _G. C. Greubel_, Jul 15 2017
%F From _Amiram Eldar_, Jan 10 2023: (Start)
%F Sum_{n>=1} 1/a(n) = zeta(3)/2.
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 3*zeta(3)/8. (End)
%p seq(2*n^3, n=0..39); # _Nathaniel Johnston_, Jun 26 2011
%t 2 Range[0, 50]^3 (* _Wesley Ivan Hurt_, Aug 25 2014 *)
%o (Magma) [2*n^3: n in [0..30]]; // _Vincenzo Librandi_, Jun 26 2011
%o (PARI) a(n)=2*n^3 \\ _Charles R Greathouse IV_, Sep 24 2015
%Y Cf. A000578, A002117, A002378, A033581, A117642, A213821.
%K nonn,easy
%O 0,2
%A _Jeff Burch_
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