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

1,1

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).

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)

Mathematica

Table[25n^2+5, {n, 50}]

Prog

(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]];
(PARI) a(n) = 25*n^2 + 5.

Crossrefs

Cf. A005843, A158187.

Sequence in context: A316362 A081370 A257891 * A046301 A193873 A266955

Adjacent sequences: A158442 A158443 A158444 * A158446 A158447 A158448

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 19 2009

Status

approved