A158593 a(n) = 38*n^2 + 1.

1, 39, 153, 343, 609, 951, 1369, 1863, 2433, 3079, 3801, 4599, 5473, 6423, 7449, 8551, 9729, 10983, 12313, 13719, 15201, 16759, 18393, 20103, 21889, 23751, 25689, 27703, 29793, 31959, 34201, 36519, 38913, 41383, 43929, 46551, 49249, 52023, 54873, 57799, 60801

Offset

0,2

Comments

The identity (38*n^2 + 1)^2 - (361*n^2 + 19)*(2*n)^2 = 1 can be written as a(n)^2 - A158592(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.: -(1 + 36*x + 39*x^2)/(x-1)^3.

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

From Amiram Eldar, Mar 14 2023: (Start)

Sum_{n>=0} 1/a(n) = (coth(Pi/sqrt(38))*Pi/sqrt(38) + 1)/2.

Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/sqrt(38))*Pi/sqrt(38) + 1)/2. (End)

Mathematica

LinearRecurrence[{3, -3, 1}, {1, 39, 153}, 50] (* Vincenzo Librandi, Feb 16 2012 *)
38*Range[0, 40]^2+1 (* Harvey P. Dale, Apr 15 2019 *)

Prog

(Magma) I:=[1, 39, 153]; [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 16 2012
(PARI) for(n=0, 40, print1(38*n^2 + 1", ")); \\ Vincenzo Librandi, Feb 16 2012

Crossrefs

Cf. A005843, A158592.

Sequence in context: A072253 A128826 A240902 * A158598 A105838 A251335

Adjacent sequences: A158590 A158591 A158592 * A158594 A158595 A158596

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 22 2009

Extensions

Comment rewritten, formula replaced by R. J. Mathar, Oct 28 2009

Status

approved