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%I #57 Feb 22 2024 10:58:20
%S 0,16,144,576,1600,3600,7056,12544,20736,32400,48400,69696,97344,
%T 132496,176400,230400,295936,374544,467856,577600,705600,853776,
%U 1024144,1218816,1440000,1690000,1971216,2286144,2637376,3027600
%N a(n) = (2n(n+1))^2.
%C Arises from middle column of following triangle: 4^2, 12^2, 24^2,...:
%C ....................... 3^2 + 4^2 = 5^2
%C ............... 10^2 + 11^2 + 12^2 = 13^2 + 14^2
%C ........ 21^2 + 22^2 + 23^2 + 24^2 = 25^2 + 26^2 + 27^2
%C . 36^2 + 37^2 + 38^2 + 39^2 + 40^2 = 41^2 + 42^2 + 43^2 + 44^2, etc.
%D C. Stanley Ogilvy and John T. Anderson, Excursions in Number Theory, Oxford University Press, NY, 1966, pp. 90-92.
%H Harry J. Smith, <a href="/A060300/b060300.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5, -10, 10, -5, 1).
%F G.f.: 16*x*(1+4*x+x^2)/(1-x)^5. [_Colin Barker_, Apr 22 2012]
%F a(n) = 4*A035287(n+1) = 4*A002378(n)^2. - _Michel Marcus_, May 24 2016
%F a(n) = 16 * A000537(n) = 16 * (n*(n+1)/2)^2 = 16 * A000217(n)^2 = A046092(n)^2. - _Bruce J. Nicholson_, Jun 05 2017
%F a(n) = Integral_{x=1..2*n+1} (x^3-x) dx. - _César Aguilera_, Jun 27 2020
%t CoefficientList[Series[16 x (1 + 4 x + x^2) / (1 - x)^5, {x, 0, 33}], x] (* _Vincenzo Librandi_, Nov 18 2016 *)
%t Table[(2n(n+1))^2,{n,0,30}] (* _Harvey P. Dale_, Jan 19 2019 *)
%o (PARI) { for (n=0, 1000, write("b060300.txt", n, " ", (2*n*(n + 1))^2); ) } \\ _Harry J. Smith_, Jul 03 2009
%o (Magma) [(2*n*(n+1))^2: n in [0..30]]; // _Vincenzo Librandi_, Nov 18 2016
%Y Cf. A000217, A000537, A002378, A035287, A046092.
%K nonn,easy
%O 0,2
%A _Jason Earls_, Mar 25 2001
%E Corrected the definition from 2n(n+1)^2 to (2n(n+1))^2. - _Harry J. Smith_, Jul 03 2009
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