A157330 a(n) = 64*n - 8.

56, 120, 184, 248, 312, 376, 440, 504, 568, 632, 696, 760, 824, 888, 952, 1016, 1080, 1144, 1208, 1272, 1336, 1400, 1464, 1528, 1592, 1656, 1720, 1784, 1848, 1912, 1976, 2040, 2104, 2168, 2232, 2296, 2360, 2424, 2488, 2552, 2616, 2680, 2744, 2808, 2872, 2936, 3000

Offset: 1

Author

Vincenzo Librandi, Feb 27 2009

Keywords

• nonn
• easy

Comments

The identity (128*n^2 - 32*n + 1)^2 - (4*n^2 - n)*(64*n - 8)^2 = 1 can be written as A157331(n)^2 - A033991(n)*a(n)^2 = 1. This is the case s=2 of the identity (8*n^2*s^4 - 8*n*s^2 + 1)^2 - (n^2*s^2 - n)*(8*n*s^3 - 4*s)^2 = 1. - Vincenzo Librandi, Jan 29 2012

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..10000
• Vincenzo Librandi, X^2-AY^2=1.
• Index entries for linear recurrences with constant coefficients, signature (2,-1).

Crossrefs

Cf. A004771, A033991, A157331.

Programs

Magma

I:=[56, 120]; [n le 2 select I[n] else 2*Self(n-1)-Self(n-2): n in [1..50]]; // Vincenzo Librandi, Jan 29 2012

Mathematica

LinearRecurrence[{2,-1},{56,120},50] (* Vincenzo Librandi, Jan 29 2012 *)
64 Range[50]-8 (* Harvey P. Dale, Dec 31 2024 *)

PARI

for(n=1, 40, print1(64*n - 8", ")); \\ Vincenzo Librandi, Jan 29 2012

Formula

From Vincenzo Librandi, Jan 29 2012: (Start)

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

G.f.: x*(8*x+56)/(x-1)^2. (End)

a(n) = 8*A004771(n-1). - Michel Marcus, Aug 19 2018

E.g.f.: 8*(exp(x)*(8*x - 1) + 1). - Elmo R. Oliveira, Apr 04 2025