A158744 a(n) = 74*n^2 - 1.

73, 295, 665, 1183, 1849, 2663, 3625, 4735, 5993, 7399, 8953, 10655, 12505, 14503, 16649, 18943, 21385, 23975, 26713, 29599, 32633, 35815, 39145, 42623, 46249, 50023, 53945, 58015, 62233, 66599, 71113, 75775, 80585, 85543, 90649, 95903, 101305, 106855, 112553

Offset

1,1

Comments

The identity (74*n^2 - 1)^2 - (1369*n^2 - 37)*(2*n)^2 = 1 can be written as a(n)^2 - A158743(n)*A005843(n)^2 = 1.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..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.: x*(-73 - 76*x + 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>=1} 1/a(n) = (1 - cot(Pi/sqrt(74))*Pi/sqrt(74))/2.

Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/sqrt(74))*Pi/sqrt(74) - 1)/2. (End)

Mathematica

LinearRecurrence[{3, -3, 1}, {73, 295, 665}, 50] (* Vincenzo Librandi, Feb 21 2012 *)

Prog

(Magma) I:=[73, 295, 665]; [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=1, 40, print1(58*n^2 + 1", ")); \\ _Vincenzo Librand_i, Feb 21 2012

Crossrefs

Cf. A005843, A158743.

Sequence in context: A158740 A174334 A142614 * A297430 A142406 A005108

Adjacent sequences: A158741 A158742 A158743 * A158745 A158746 A158747

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 25 2009

Extensions

Comment rewritten and formula replaced by R. J. Mathar, Oct 22 2009

Status

approved