A158588 -id:A158588 - cp's OEIS frontend

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

Vincenzo Librandi, Mar 22 2009

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.

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).

Cf. A005843, A158588, A321180.

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

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

for(n=1, 50, print1(289*n^2-17", ")); \\ Vincenzo Librandi, Feb 15 2012

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)

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

A158729 a(n) = 1156*n^2 - 34.

Original entry on oeis.org

1122, 4590, 10370, 18462, 28866, 41582, 56610, 73950, 93602, 115566, 139842, 166430, 195330, 226542, 260066, 295902, 334050, 374510, 417282, 462366, 509762, 559470, 611490, 665822, 722466, 781422, 842690, 906270, 972162, 1040366, 1110882, 1183710, 1258850, 1336302

Offset: 1

Vincenzo Librandi, Mar 25 2009

The identity (68*n^2 - 1)^2 - (1156*n^2 - 34)*(2*n)^2 = 1 can be written as A158730(n)^2 - a(n)*A005843(n)^2 = 1.

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).

Cf. A005843, A158588, A158730.

I:=[1122, 4590, 10370]; [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 20 2012

LinearRecurrence[{3, -3, 1}, {1122, 4590, 10370}, 50] (* Vincenzo Librandi, Feb 20 2012 *)

for(n=1, 40, print1(1156*n^2 - 34", ")); \\ Vincenzo Librandi, Feb 20 2012

G.f.: 34*x*(-33 - 36*x + x^2)/(x-1)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
From Amiram Eldar, Mar 22 2023: (Start)
Sum_{n>=1} 1/a(n) = (1 - cot(Pi/sqrt(34))*Pi/sqrt(34))/68.
Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/sqrt(34))*Pi/sqrt(34) - 1)/68. (End)
From Elmo R. Oliveira, Jan 16 2025: (Start)
E.g.f.: 34*(exp(x)*(34*x^2 + 34*x - 1) + 1).
a(n) = 34*A158588(n). (End)

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

cp's OEIS Frontend

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

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