A158562 -id:A158562 - cp's OEIS frontend

31, 127, 287, 511, 799, 1151, 1567, 2047, 2591, 3199, 3871, 4607, 5407, 6271, 7199, 8191, 9247, 10367, 11551, 12799, 14111, 15487, 16927, 18431, 19999, 21631, 23327, 25087, 26911, 28799, 30751, 32767, 34847, 36991, 39199, 41471, 43807, 46207, 48671, 51199, 53791

Offset: 1

Vincenzo Librandi, Mar 21 2009

The identity (32*n^2 - 1)^2 - (256*n^2 - 16)*(2*n)^2 = 1 can be written as a(n)^2 - A158562(n)*A005843(n)^2 = 1. [comment rewritten by R. J. Mathar, Oct 16 2009]
From Omar E. Pol, Apr 21 2021: (Start)
Sequence found by reading the line from 31, in the direction 31, 127, ..., in the rectangular spiral whose vertices are the generalized 18-gonal numbers A274979.
The spiral begins as follows:
       46_   _   _   _   _   _   _   _   _   _18
        |                                                        |
        |                            0                           |
        |                            |  _   _   _   _  |
        |                            1                           15
        |
       51
(End)

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
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, A010021, A158562, A158575, A244082.

Cf. A274979 (generalized 18-gonal numbers).

[32*n^2-1: n in [1..40]]; // Vincenzo Librandi, Sep 11 2013

32 Range[40]^2 - 1  (* Harvey P. Dale, Mar 04 2011 *)
CoefficientList[Series[(- 31 - 34 x + x^2) / (x - 1)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Sep 11 2013 *)

a(n)=32*n^2-1 \\ Charles R Greathouse IV, Jun 17 2017

G.f.: x*(-31-34*x+x^2)/(x-1)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = A244082(n) - 1. - Omar E. Pol, Apr 21 2021
From Amiram Eldar, Mar 09 2023: (Start)
Sum_{n>=1} 1/a(n) = (1 - cot(Pi/(4*sqrt(2)))*Pi/(4*sqrt(2)))/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/(4*sqrt(2)))*Pi/(4*sqrt(2)) - 1)/2. (End)
E.g.f.: 1 + exp(x)*(32*x^2 + 32*x - 1). - Elmo R. Oliveira, Jan 25 2025

cp's OEIS Frontend

A158563 a(n) = 32*n^2 - 1.

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