A132592 X-values of solutions to the equation X*(X + 1) - 8*Y^2 = 0.

0, 8, 288, 9800, 332928, 11309768, 384199200, 13051463048, 443365544448, 15061377048200, 511643454094368, 17380816062160328, 590436102659356800, 20057446674355970888, 681362750825443653408, 23146276081390728245000, 786292024016459316676608, 26710782540478226038759688

Offset: 0

Author

Mohamed Bouhamida, Nov 14 2007

Keywords

• nonn
• easy

Comments

Equivalently, numbers k such that both k/2 and k+1 are squares. - Karl-Heinz Hofmann, Sep 20 2022

Equivalently, numbers k such that the k-dimensional volume and total (k-1)-dimensional volume are equal, with side length being a positive integer, for all regular polyhedra constructible in k dimensions. - Matt Moir, Jul 09 2024

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..500
• Index entries for linear recurrences with constant coefficients, signature (35,-35,1).

Crossrefs

Intersection between A132411 and A001105.

Cf. A007654.

Cf. A001541, A058331, A001079, A037270, A055792, A071253, A108741, A132592, A146311, A146312, A146313, A173115, A173116, A173121.

Programs

Magma

I:=[0,8,288]; [n le 3 select I[n] else 35*Self(n-1)-35*Self(n-2)+ Self(n-3): n in [1..30]]; // Vincenzo Librandi, Dec 24 2018

Mathematica

Table[Round[N[Sinh[2 n ArcCosh[Sqrt[2]]]^2, 100]], {n, 0, 20}] (* Artur Jasinski, Feb 10 2010 *)
LinearRecurrence[{35, -35, 1}, {0, 8, 288}, 30] (* Vincenzo Librandi, Dec 24 2018 *)

Python

A132592 = [0, 8]
for n in range(2, 18): A132592.append(34 * A132592[-1] - A132592[-2] + 16)
print(A132592) # Karl-Heinz Hofmann, Sep 20 2022

Formula

a(0)=0, a(1)=8 and a(n) = 34*a(n-1) - a(n-2) + 16.

a(n) = (A056771(n) - 1)/2. - Max Alekseyev, Nov 13 2009

a(n) = sinh(2*n*arccosh(sqrt(2))^2) (n=0,1,2,3,...). - Artur Jasinski, Feb 10 2010

G.f.: -8*x*(x+1)/((x-1)*(x^2-34*x+1)). - Colin Barker, Oct 24 2012

a(n) = A055792(n+1)-1 = A001541(n)^2 - 1. - Antti Karttunen, Oct 03 2016

Extensions

More terms from Max Alekseyev, Nov 13 2009