A158743 a(n) = 1369*n^2 - 37.

1332, 5439, 12284, 21867, 34188, 49247, 67044, 87579, 110852, 136863, 165612, 197099, 231324, 268287, 307988, 350427, 395604, 443519, 494172, 547563, 603692, 662559, 724164, 788507, 855588, 925407, 997964, 1073259, 1151292, 1232063, 1315572, 1401819, 1490804, 1582527

Offset

1,1

Comments

The identity (74*n^2 - 1)^2 - (1369*n^2 - 37)*(2*n)^2 = 1 can be written as A158744(n)^2 - a(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.: 37*x*(-36 - 39*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(37))*Pi/sqrt(37))/74.

Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/sqrt(37))*Pi/sqrt(37) - 1)/74. (End)

Mathematica

LinearRecurrence[{3, -3, 1}, {1332, 5439, 12284}, 50] (* Vincenzo Librandi, Feb 21 2012 *)
1369*Range[30]^2-37 (* Harvey P. Dale, Aug 12 2021 *)

Prog

(Magma) I:=[1332, 5439, 12284]; [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(1369*n^2 - 37", ")); \\ Vincenzo Librandi, Feb 21 2012

Crossrefs

Cf. A005843, A158744.

Sequence in context: A052072 A050646 A044882 * A188308 A185494 A035862

Adjacent sequences: A158740 A158741 A158742 * A158744 A158745 A158746

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 25 2009

Extensions

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

Status

approved