A158668 a(n) = 58*n^2 - 1.

57, 231, 521, 927, 1449, 2087, 2841, 3711, 4697, 5799, 7017, 8351, 9801, 11367, 13049, 14847, 16761, 18791, 20937, 23199, 25577, 28071, 30681, 33407, 36249, 39207, 42281, 45471, 48777, 52199, 55737, 59391, 63161, 67047, 71049, 75167, 79401, 83751, 88217, 92799

Offset

1,1

Comments

The identity (58*n^2 - 1)^2 - (841*n^2 - 29)*(2*n)^2 = 1 can be written as a(n)^2 - A158667(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*(-57 - 60*x + 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>=1} 1/a(n) = (1 - cot(Pi/sqrt(58))*Pi/sqrt(58))/2.

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

Mathematica

LinearRecurrence[{3, -3, 1}, {57, 231, 521}, 50] (* Vincenzo Librandi, Feb 18 2012 *)

Prog

(Magma) I:=[57, 231, 521]; [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=1, 40, print1(58*n^2 - 1", ")); \\ Vincenzo Librandi, Feb 18 2012

Crossrefs

Cf. A005843, A158667.

Sequence in context: A277805 A158660 A358332 * A145296 A176635 A048422

Adjacent sequences: A158665 A158666 A158667 * A158669 A158670 A158671

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