A158767 a(n) = 76*n^2 + 1.

1, 77, 305, 685, 1217, 1901, 2737, 3725, 4865, 6157, 7601, 9197, 10945, 12845, 14897, 17101, 19457, 21965, 24625, 27437, 30401, 33517, 36785, 40205, 43777, 47501, 51377, 55405, 59585, 63917, 68401, 73037, 77825, 82765, 87857, 93101, 98497, 104045, 109745, 115597

Offset

0,2

Comments

The identity (76*n^2 + 1)^2 - (1444*n^2 + 38)*(2*n)^2 = 1 can be written as a(n)^2 - A158766(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 + 74*x + 77*x^2)/(x-1)^3.

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

From Amiram Eldar, Mar 23 2023: (Start)

Sum_{n>=0} 1/a(n) = (coth(Pi/(2*sqrt(19)))*Pi/(2*sqrt(19)) + 1)/2.

Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/(2*sqrt(19)))*Pi/(2*sqrt(19)) + 1)/2. (End)

Mathematica

LinearRecurrence[{3, -3, 1}, {1, 77, 305}, 50] (* Vincenzo Librandi, Feb 21 2012 *)
76*Range[0, 40]^2+1 (* Harvey P. Dale, Jan 19 2016 *)

Prog

(Magma) I:=[1, 77, 305]; [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 21 2012
(PARI) for(n=0, 40, print1(76*n^2 + 1", ")); \\ Vincenzo Librandi, Feb 21 2012

Crossrefs

Cf. A005843, A158766.

Sequence in context: A029558 A156652 A298102 * A158771 A331976 A020206

Adjacent sequences: A158764 A158765 A158766 * A158768 A158769 A158770

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 26 2009

Extensions

Comment rewritten, a(0) added, and formula replaced by R. J. Mathar, Oct 22 2009

Status

approved