A158632 a(n) = 46*n^2 + 1.

1, 47, 185, 415, 737, 1151, 1657, 2255, 2945, 3727, 4601, 5567, 6625, 7775, 9017, 10351, 11777, 13295, 14905, 16607, 18401, 20287, 22265, 24335, 26497, 28751, 31097, 33535, 36065, 38687, 41401, 44207, 47105, 50095, 53177, 56351, 59617, 62975, 66425, 69967, 73601

Offset

0,2

Comments

The identity (46*n^2 + 1)^2 - (529*n^2 + 23)*(2*n)^2 = 1 can be written as a(n)^2 - A158631(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

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

G.f.: -(1 + 44*x + 47*x^2)/(x-1)^3.

From Amiram Eldar, Mar 16 2023: (Start)

Sum_{n>=0} 1/a(n) = (coth(Pi/sqrt(46))*Pi/sqrt(46) + 1)/2.

Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/sqrt(46))*Pi/sqrt(46) + 1)/2. (End)

Mathematica

LinearRecurrence[{3, -3, 1}, {1, 47, 185}, 50] (* Vincenzo Librandi, Feb 17 2012 *)
46*Range[0, 40]^2+1 (* Harvey P. Dale, Mar 28 2020 *)

Prog

(Magma) I:=[1, 47, 185]; [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(46*n^2 + 1", ")); \\ Vincenzo Librandi, Feb 17 2012

Crossrefs

Cf. A005843, A158631.

Sequence in context: A253344 A227282 A204610 * A142413 A065532 A157362

Adjacent sequences: A158629 A158630 A158631 * A158633 A158634 A158635

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 23 2009

Extensions

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

Status

approved