A158588 a(n) = 34*n^2 - 1.

33, 135, 305, 543, 849, 1223, 1665, 2175, 2753, 3399, 4113, 4895, 5745, 6663, 7649, 8703, 9825, 11015, 12273, 13599, 14993, 16455, 17985, 19583, 21249, 22983, 24785, 26655, 28593, 30599, 32673, 34815, 37025, 39303, 41649, 44063, 46545, 49095, 51713, 54399, 57153

Offset

1,1

Comments

The identity (34*n^2 - 1)^2 - (289*n^2 - 17)*(2*n)^2 = 1 can be written as a(n)^2 - A158587(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.: x*(-33 - 36*x + x^2)/(x-1)^3.

From Amiram Eldar, Mar 14 2023: (Start)

Sum_{n>=1} 1/a(n) = (1 - cot(Pi/sqrt(34))*Pi/sqrt(34))/2.

Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/sqrt(34))*Pi/sqrt(34) - 1)/2. (End)

Mathematica

LinearRecurrence[{3, -3, 1}, {33, 135, 305}, 50] (* Vincenzo Librandi, Feb 16 2012 *)
34*Range[40]^2-1 (* Harvey P. Dale, Feb 10 2015 *)

Prog

(Magma) I:=[33, 135, 305]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..50]]; // Vincenzo Librandi, Feb 16 2012
(PARI) for(n=1, 40, print1(34*n^2-1", ")); \\ Vincenzo Librandi, Feb 16 2012

Crossrefs

Cf. A005843, A158587.

Sequence in context: A044365 A044746 A105091 * A232538 A231758 A215962

Adjacent sequences: A158585 A158586 A158587 * A158589 A158590 A158591

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 22 2009

Extensions

Comment rewritten by R. J. Mathar, Oct 16 2009

Status

approved