A158445 a(n) = 25*n^2 + 5.

30, 105, 230, 405, 630, 905, 1230, 1605, 2030, 2505, 3030, 3605, 4230, 4905, 5630, 6405, 7230, 8105, 9030, 10005, 11030, 12105, 13230, 14405, 15630, 16905, 18230, 19605, 21030, 22505, 24030, 25605, 27230, 28905, 30630, 32405, 34230, 36105, 38030, 40005, 42030

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 A158187(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, A158187, A212656.

Programs

Magma

I:=[30, 105, 230]; [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},{30,105,230},50] (* Harvey P. Dale, Mar 21 2025 *)

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*(6+3*x+x^2)/(1-x)^3.

From Amiram Eldar, Mar 05 2023: (Start)

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

Sum_{n>=1} (-1)^(n+1)/a(n) = (1 - cosech(Pi/sqrt(5))*Pi/sqrt(5))/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*A212656(n). (End)