A158776 a(n) = 80*n^2 + 1.

1, 81, 321, 721, 1281, 2001, 2881, 3921, 5121, 6481, 8001, 9681, 11521, 13521, 15681, 18001, 20481, 23121, 25921, 28881, 32001, 35281, 38721, 42321, 46081, 50001, 54081, 58321, 62721, 67281, 72001, 76881, 81921, 87121, 92481, 98001, 103681, 109521, 115521, 121681

Offset: 0

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 a(n)^2 - A158775(n)*A005843(n)^2 = 1.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..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, A158493, A158775.

Programs

Magma

I:=[1, 81, 321]; [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}, {1, 81, 321}, 50] (* Vincenzo Librandi, Feb 20 2012 *)
80 Range[0,50]^2+1 (* Harvey P. Dale, Jan 17 2021 *)

PARI

a(n)=80*n^2+1 \\ Charles R Greathouse IV, Dec 23 2011

Formula

G.f.: -(1 + 78*x + 81*x^2)/(x-1)^3.

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

From Amiram Eldar, Mar 24 2023: (Start)

Sum_{n>=0} 1/a(n) = (coth(Pi/(4*sqrt(5)))*Pi/(4*sqrt(5)) + 1)/2.

Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/(4*sqrt(5)))*Pi/(4*sqrt(5)) + 1)/2. (End)

From Elmo R. Oliveira, Jan 25 2025: (Start)

E.g.f.: exp(x)*(1 + 80*x + 80*x^2).

a(n) = A158493(2*n). (End)

Extensions

Comment rewritten, a(0) added and formula replaced by R. J. Mathar, Oct 22 2009