A158666 a(n) = 58*n^2 + 1.

1, 59, 233, 523, 929, 1451, 2089, 2843, 3713, 4699, 5801, 7019, 8353, 9803, 11369, 13051, 14849, 16763, 18793, 20939, 23201, 25579, 28073, 30683, 33409, 36251, 39209, 42283, 45473, 48779, 52201, 55739, 59393, 63163, 67049, 71051, 75169, 79403, 83753, 88219, 92801

Offset

0,2

Comments

The identity (58*n^2 + 1)^2 - (841*n^2 + 29)*(2*n)^2 = 1 can be written as a(n)^2 - A158665(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 + 56*x + 59*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(58))*Pi/sqrt(58) + 1)/2.

Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/sqrt(58))*Pi/sqrt(58) + 1)/2. (End)

Mathematica

LinearRecurrence[{3, -3, 1}, {1, 59, 233}, 50] (* Vincenzo Librandi, Feb 18 2012 *)

Prog

(Magma) I:=[1, 59, 233]; [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 18 2012
(PARI) for(n=0, 40, print1(58*n^2 + 1", ")); \\ Vincenzo Librandi, Feb 18 2012

Crossrefs

Cf. A005843, A158665.

Sequence in context: A142215 A141977 A059256 * A060331 A158670 A142046

Adjacent sequences: A158663 A158664 A158665 * A158667 A158668 A158669

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 24 2009

Extensions

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

Status

approved