A158446 a(n) = 25*n^2 - 5.

20, 95, 220, 395, 620, 895, 1220, 1595, 2020, 2495, 3020, 3595, 4220, 4895, 5620, 6395, 7220, 8095, 9020, 9995, 11020, 12095, 13220, 14395, 15620, 16895, 18220, 19595, 21020, 22495, 24020, 25595, 27220, 28895, 30620, 32395, 34220, 36095, 38020, 39995, 42020, 44095

Offset: 1

Author

Vincenzo Librandi, Mar 19 2009

Keywords

• nonn
• easy

Comments

The identity (10*n^2 - 1)^2 - (25*n^2 - 5)*(2*n)^2 = 1 can be written as A158447(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, A134538, A158447.

Programs

Magma

I:=[20, 95, 220]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]];

Mathematica

Table[25n^2-5,{n,50}]
LinearRecurrence[{3,-3,1},{20,95,220},40] (* Harvey P. Dale, May 05 2019 *)

PARI

a(n) = 25*n^2 - 5.

Formula

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

G.f: 5*x*(4+7*x-x^2)/(1-x)^3.

From Amiram Eldar, Mar 05 2023: (Start)

Sum_{n>=1} 1/a(n) = (1 - cot(Pi/sqrt(5))*Pi/sqrt(5))/10.

Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/sqrt(5))*Pi/sqrt(5) - 1)/10. (End)

From Elmo R. Oliveira, Jan 16 2025: (Start)

E.g.f.: 5*(exp(x)*(5*x^2 + 5*x - 1) + 1).

a(n) = 5*A134538(n). (End)