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

0, 8, 80, 360, 1088, 2600, 5328, 9800, 16640, 26568, 40400, 59048, 83520, 114920, 154448, 203400, 263168, 335240, 421200, 522728, 641600, 779688, 938960, 1121480, 1329408, 1565000, 1830608, 2128680, 2461760, 2832488, 3243600

Offset: 0

Author

Artur Jasinski, Feb 10 2010

Keywords

• nonn
• easy

Examples

			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
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A001079, A037270, A071253, A108741, A132592, A146311, A146312, A146313, A173115, A173121.

Programs

Magma

[4*n^2*(n^2+1): n in [0..40]]; // Vincenzo Librandi, Jun 15 2011

Mathematica

Table[4*n^2*(n^2 + 1), {n, 0, 30}] (* OR *)
Table[Round[N[Sinh[2 ArcSinh[n]]^2, 100]], {n, 0, 30}]
a[ n_] := TrigExpand @ Sinh[ 2 ArcSinh @ n]^2; (* Michael Somos, Jul 05 2018 *)

PARI

a(n)=4*n^2*(n^2+1) \\ Charles R Greathouse IV, Apr 17 2012

PARI

a(n)=8*binomial(n^2+1,2) \\ Charles R Greathouse IV, Apr 17 2012

Formula

a(n) = 4*A071253(n) = 8*A037270(n).

G.f.: 8*x*(1 + 5*x + 5*x^2 + x^3)/(1 - x)^5. - Colin Barker, Jan 08 2012

E.g.f.: 4*x*(2 + 8*x + 6*x^2 + x^3)*exp(x). - Michael Somos, Jul 05 2018

a(n) = a(-n) = (2*n)^2 + (2*n^2)^2 = (2*n^2 + 1)^2 - 1. - Michael Somos, Jul 05 2018

From Amiram Eldar, Oct 25 2024: (Start)

Sum_{n>=1} 1/a(n) = Pi^2/24 + (1-Pi*coth(Pi))/8.

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/48 + (Pi*cosech(Pi)-1)/8. (End)

Extensions

Name corrected by Jianing Song, Nov 23 2018