A158587 a(n) = 289*n^2 - 17.

272, 1139, 2584, 4607, 7208, 10387, 14144, 18479, 23392, 28883, 34952, 41599, 48824, 56627, 65008, 73967, 83504, 93619, 104312, 115583, 127432, 139859, 152864, 166447, 180608, 195347, 210664, 226559, 243032, 260083, 277712, 295919, 314704, 334067, 354008, 374527

Offset

1,1

Comments

The identity (34*n^2 - 1)^2 - (289*n^2 - 17)*(2*n)^2 = 1 can be written as A158588(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

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

G.f.: 17*x*(-16 - 19*x + x^2)/(x-1)^3.

From Amiram Eldar, Mar 14 2023: (Start)

Sum_{n>=1} 1/a(n) = (1 - cot(Pi/sqrt(17))*Pi/sqrt(17))/34.

Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/sqrt(17))*Pi/sqrt(17) - 1)/34. (End)

From Elmo R. Oliveira, Jan 16 2025: (Start)

E.g.f.: 17*(exp(x)*(17*x^2 + 17*x - 1) + 1).

a(n) = 17*A321180(n). (End)

Mathematica

LinearRecurrence[{3, -3, 1}, {272, 1139, 2584}, 50] (* Vincenzo Librandi, Feb 15 2012 *)
289*Range[40]^2-17 (* Harvey P. Dale, Jan 30 2019 *)

Prog

(Magma) I:=[272, 1139, 2584]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..50]]; // Vincenzo Librandi, Feb 15 2012
(PARI) for(n=1, 50, print1(289*n^2-17", ")); \\ Vincenzo Librandi, Feb 15 2012

Crossrefs

Cf. A005843, A158588, A321180.

Sequence in context: A317269 A253113 A234884 * A304417 A316242 A230908

Adjacent sequences: A158584 A158585 A158586 * A158588 A158589 A158590

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 22 2009

Extensions

Comment rewritten by R. J. Mathar, Oct 16 2009

Status

approved