A158775 a(n) = 1600*n^2 + 40.

1640, 6440, 14440, 25640, 40040, 57640, 78440, 102440, 129640, 160040, 193640, 230440, 270440, 313640, 360040, 409640, 462440, 518440, 577640, 640040, 705640, 774440, 846440, 921640, 1000040, 1081640, 1166440, 1254440, 1345640, 1440040, 1537640, 1638440, 1742440

Offset: 1

Author

Vincenzo Librandi, Mar 26 2009

Keywords

• nonn
• easy

Comments

The identity (80*n^2 + 1)^2 - (1600*n^2 + 40)*(2*n)^2 = 1 can be written as A158776(n)^2 - a(n)*A005843(n)^2 = 1.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..10000
• Vincenzo Librandi, X^2-AY^2=1, Math Forum, 2007. [Wayback Machine link]
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A005843, A158602, A158776.

Programs

Magma

I:=[1640, 6440, 14440]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 20 2012

Mathematica

LinearRecurrence[{3, -3, 1}, {1640, 6440, 14440}, 50] (* Vincenzo Librandi, Feb 20 2012 *)
1600*Range[30]^2+ 40 (* Harvey P. Dale, Jun 02 2020 *)

PARI

for(n=1, 40, print1(1600*n^2 + 40", ")); \\ Vincenzo Librandi, Feb 20 2012

Formula

From R. J. Mathar, Jul 26 2009: (Start)

G.f.: -40*x*(41 + 38*x + x^2)/(x-1)^3.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)

From Amiram Eldar, Mar 24 2023: (Start)

Sum_{n>=1} 1/a(n) = (coth(Pi/(2*sqrt(10)))*Pi/(2*sqrt(10)) - 1)/80.

Sum_{n>=1} (-1)^(n+1)/a(n) = (1 - cosech(Pi/(2*sqrt(10)))*Pi/(2*sqrt(10)))/80. (End)

From Elmo R. Oliveira, Jan 26 2025: (Start)

E.g.f.: 40*(exp(x)*(40*x^2 + 40*x + 1) - 1).

a(n) = 40*A158602(n). (End)

Extensions

Edited by R. J. Mathar, Jul 26 2009