A158665 a(n) = 841*n^2 + 29.

29, 870, 3393, 7598, 13485, 21054, 30305, 41238, 53853, 68150, 84129, 101790, 121133, 142158, 164865, 189254, 215325, 243078, 272513, 303630, 336429, 370910, 407073, 444918, 484445, 525654, 568545, 613118, 659373, 707310, 756929, 808230, 861213, 915878, 972225

Offset

0,1

Comments

The identity (58*n^2 + 1)^2 - (841*n^2 + 29)*(2*n)^2 = 1 can be written as A158666(n)^2 - a(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).

Formula

G.f.: -29*(1 + 27*x + 30*x^2)/(x-1)^3.

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

From Amiram Eldar, Mar 20 2023: (Start)

Sum_{n>=0} 1/a(n) = (coth(Pi/sqrt(29))*Pi/sqrt(29) + 1)/58.

Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/sqrt(29))*Pi/sqrt(29) + 1)/58. (End)

E.g.f.: 29*exp(x)*(1 + 29*x + 29*x^2). - Elmo R. Oliveira, Jan 13 2025

Mathematica

29(29Range[0, 40]^2+1) (* or *) LinearRecurrence[{3, -3, 1}, {29, 870, 3393}, 40] (* Harvey P. Dale, Nov 05 2011 *)

Prog

(Magma) I:=[29, 870, 3393]; [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 17 2012
(PARI) for(n=0, 40, print1(841*n^2 + 29", ")); \\ Vincenzo Librandi, Feb 17 2012

Crossrefs

Cf. A005843, A158666.

Sequence in context: A049667 A042626 A157877 * A324432 A167738 A107964

Adjacent sequences: A158662 A158663 A158664 * A158666 A158667 A158668

Keyword

nonn,easy,changed

Author

Vincenzo Librandi, Mar 24 2009

Extensions

Comment rephrased and redundant formula replaced by R. J. Mathar, Oct 19 2009

Status

approved