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A173116 a(n) = sinh(2*arcsinh(n))^2 = 4*n^2*(n^2 + 1).

%I A173116 #36 Oct 25 2024 07:24:36
%S A173116 0,8,80,360,1088,2600,5328,9800,16640,26568,40400,59048,83520,114920,
%T A173116 154448,203400,263168,335240,421200,522728,641600,779688,938960,
%U A173116 1121480,1329408,1565000,1830608,2128680,2461760,2832488,3243600
%N A173116 a(n) = sinh(2*arcsinh(n))^2 = 4*n^2*(n^2 + 1).
%H A173116 Vincenzo Librandi, <a href="/A173116/b173116.txt">Table of n, a(n) for n = 0..1000</a>
%H A173116 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A173116 a(n) = 4*A071253(n) = 8*A037270(n).
%F A173116 G.f.: 8*x*(1 + 5*x + 5*x^2 + x^3)/(1 - x)^5. - _Colin Barker_, Jan 08 2012
%F A173116 E.g.f.: 4*x*(2 + 8*x + 6*x^2 + x^3)*exp(x). - _Michael Somos_, Jul 05 2018
%F A173116 a(n) = a(-n) = (2*n)^2 + (2*n^2)^2 = (2*n^2 + 1)^2 - 1. - _Michael Somos_, Jul 05 2018
%F A173116 From _Amiram Eldar_, Oct 25 2024: (Start)
%F A173116 Sum_{n>=1} 1/a(n) = Pi^2/24 + (1-Pi*coth(Pi))/8.
%F A173116 Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/48 + (Pi*cosech(Pi)-1)/8. (End)
%e A173116 G.f. = 8*x + 80*x^2 + 360*x^3 + 1088*x^4 + 2600*x^5 + 5328*x^6 + 9800*x^7 + ... - _Michael Somos_, Jul 05 2018
%t A173116 Table[4*n^2*(n^2 + 1), {n, 0, 30}] (* OR *)
%t A173116 Table[Round[N[Sinh[2 ArcSinh[n]]^2, 100]], {n, 0, 30}]
%t A173116 a[ n_] := TrigExpand @ Sinh[ 2 ArcSinh @ n]^2; (* _Michael Somos_, Jul 05 2018 *)
%o A173116 (Magma) [4*n^2*(n^2+1): n in [0..40]]; // _Vincenzo Librandi_, Jun 15 2011
%o A173116 (PARI) a(n)=4*n^2*(n^2+1) \\ _Charles R Greathouse IV_, Apr 17 2012
%o A173116 (PARI) a(n)=8*binomial(n^2+1,2) \\ _Charles R Greathouse IV_, Apr 17 2012
%Y A173116 Cf. A001079, A037270, A071253, A108741, A132592, A146311, A146312, A146313, A173115, A173121.
%K A173116 nonn,easy
%O A173116 0,2
%A A173116 _Artur Jasinski_, Feb 10 2010
%E A173116 Name corrected by _Jianing Song_, Nov 23 2018