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

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

Author

Vincenzo Librandi, Mar 21 2009

Keywords

• nonn
• easy

Comments

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)

Links

• 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).

Crossrefs

Cf. A005843, A010021, A158562, A158575, A244082.

Cf. A274979 (generalized 18-gonal numbers).

Programs

Magma

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

Mathematica

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 *)

PARI

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

Formula

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