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

0,2

Comments

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

Links

Seiichi Manyama, Table of n, a(n) for n = 0..500

Index entries for linear recurrences with constant coefficients, signature (35,-35,1).

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

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 *)

Prog

(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
(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

Crossrefs

Intersection between A132411 and A001105.

Cf. A007654.

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

Sequence in context: A187191 A054607 A254408 * A034977 A065141 A190840

Adjacent sequences: A132589 A132590 A132591 * A132593 A132594 A132595

Keyword

nonn,easy

Author

Mohamed Bouhamida, Nov 14 2007

Extensions

More terms from Max Alekseyev, Nov 13 2009

Status

approved