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%I #50 Apr 16 2023 21:10:22
%S 0,2,20,90,272,650,1332,2450,4160,6642,10100,14762,20880,28730,38612,
%T 50850,65792,83810,105300,130682,160400,194922,234740,280370,332352,
%U 391250,457652,532170,615440,708122,810900,924482,1049600,1187010,1337492,1501850,1680912
%N a(n) = n^2*(n^2+1).
%C The identity (n^5+n^3)^2+(n^2*(n^2+1))^2 = n*(n^3+n)^3 can be written as A155977(n)^2+a(n)^2 = n*A034262(n)^3. - _Vincenzo Librandi_, Aug 08 2010
%H Vincenzo Librandi, <a href="/A071253/b071253.txt">Table of n, a(n) for n = 0..1000</a>
%H T. A. Gulliver, <a href="https://web.archive.org/web/20050131200534/http://www.ece.uvic.ca/~agullive/square.ps">Sequences from Arrays of Integers</a>, Int. Math. Journal, Vol. 1, No. 4, pp. 323-332, 2002.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F a(n) = A002522(n)*A000290(n). - _Zerinvary Lajos_, Apr 20 2008
%F a(n) = (1/4)*sinh(2*arcsinh(n))^2. - _Artur Jasinski_, Feb 10 2010
%F G.f.: 2*x*(1+x)*(1+4*x+x^2)/(1-x)^5. - _Colin Barker_, Jan 08 2012
%F a(n) = A002378(A000290(n)). - _Rick L. Shepherd_, Sep 22 2014
%F Sum_{n>=1} 1/a(n) = 0.5682... = Pi^2/6- (Pi*coth Pi-1)/2 = A013661 - A259171 [J. Math. Anal. Appl. 316 (2006) 328]. - _R. J. Mathar_, Oct 18 2019
%F a(n) = 2*A037270(n). - _R. J. Mathar_, Oct 18 2019
%F Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/12 - 1/2 + Pi*cosech(Pi)/2. - _Amiram Eldar_, Nov 05 2020
%F E.g.f.: exp(x)*x*(2 + 8*x + 6*x^2 + x^3). - _Stefano Spezia_, Oct 08 2022
%F a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - _Wesley Ivan Hurt_, Apr 16 2023
%p with(combinat):seq(lcm(fibonacci(3,n),n^2),n=0..35); # _Zerinvary Lajos_, Apr 20 2008
%p a:=n->add(n+add(n+add(n, j=1..n-1),j=1..n),j=1..n):seq(a(n), n=0..21); # _Zerinvary Lajos_, Aug 27 2008
%t Table[(1/4) Round[N[Sinh[2 ArcSinh[n]]^2, 100]], {n, 0, 10}] (* _Artur Jasinski_, Feb 10 2010 *)
%t Table[n^2*(n^2+1),{n,0,80}] (* _Vladimir Joseph Stephan Orlovsky_, Apr 18 2011 *)
%t CoefficientList[Series[2 x (1 + x) (1 + 4 x + x^2)/(1 - x)^5, {x, 0, 40}], x] (* _Vincenzo Librandi_, May 29 2014 *)
%o (PARI) a(n)=n^2*(n^2+1) \\ _Charles R Greathouse IV_, Sep 24 2015
%Y Cf. A034262, A069187, A155977.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Jun 12 2002
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