A158444 a(n) = 16*n^2 + 4.

20, 68, 148, 260, 404, 580, 788, 1028, 1300, 1604, 1940, 2308, 2708, 3140, 3604, 4100, 4628, 5188, 5780, 6404, 7060, 7748, 8468, 9220, 10004, 10820, 11668, 12548, 13460, 14404, 15380, 16388, 17428, 18500, 19604, 20740, 21908, 23108, 24340, 25604, 26900, 28228

Offset: 1

Author

Vincenzo Librandi, Mar 19 2009

Keywords

• nonn
• easy

Comments

The identity (8*n^2 + 1)^2 - (16*n^2 + 4)*(2*n)^2 = 1 can be written as A081585(n)^2 - a(n)*A005843(n)^2 = 1. [rewritten by Bruno Berselli, Sep 06 2011]

Sequence found by reading the line from 20, in the direction 20, 68, ... in the square spiral whose vertices are the generalized decagonal numbers A074377. - Omar E. Pol, Nov 02 2012

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).

Crossrefs

Cf. A005843, A053755, A074377, A081585.

Programs

Magma

[16*n^2+4: n in [1..50]];

Mathematica

a[n_] := 16*n^2 + 4; Array[a, 50] (* Amiram Eldar, Mar 05 2023 *)

PARI

a(n)=16*n^2+4 \\ Charles R Greathouse IV, Jun 17 2017

Formula

From Bruno Berselli, Sep 06 2011: (Start)

G.f.: 4*x*(5 + 2*x + x^2)/(1-x)^3.

a(n) = 4*A053755(n). (End)

From Amiram Eldar, Mar 05 2023: (Start)

Sum_{n>=1} 1/a(n) = (coth(Pi/2)*Pi/2 - 1)/8.

Sum_{n>=1} (-1)^(n+1)/a(n) = (1 - cosech(Pi/2)*Pi/2)/8. (End)

E.g.f.: 4*(exp(x)*(4*x^2 + 4*x + 1) - 1). - Elmo R. Oliveira, Jan 27 2025