A132584 a(0)=0, a(1)=4; for n > 1, a(n) = 18*a(n-1) - a(n-2) + 8.

0, 4, 80, 1444, 25920, 465124, 8346320, 149768644, 2687489280, 48225038404, 865363202000, 15528312597604, 278644263554880, 5000068431390244, 89722587501469520, 1610006506595061124, 28890394531209630720, 518417095055178291844, 9302617316461999622480

Offset: 0

Author

Mohamed Bouhamida, Nov 14 2007

Keywords

• nonn
• easy

Comments

The old definition given for this sequence was "Sequence allows us to find X values of the equation: X(X + 1) - 5*Y^2 = 0".

With this old definition, if X = a(n), then Y = A207832(n). Now, with u = 2X+1, this Diophantine equation becomes the Pell-Fermat equation u^2 - 20*Y^2 = 1, and then, u = A023039(n) and Y = A207832(n). - Bernard Schott, Jan 25 2023

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..500
• Index entries for linear recurrences with constant coefficients, signature (19,-19,1).

Crossrefs

Cf. A000032, A000045, A000217, A007654, A023039, A049683, A292443, A207832.

Programs

Magma

I:=[0,4,80]; [n le 3 select I[n] else 18*Self(n-1)-Self(n-2)+8: n in [1..30]]; // Vincenzo Librandi, Dec 24 2018

Mathematica

LinearRecurrence[{19, -19, 1}, {0, 4, 80}, 40] (* Vincenzo Librandi, Dec 24 2018 *)
nxt[{a_,b_}]:={b,18b-a+8}; NestList[nxt,{0,4},20][[;;,1]] (* Harvey P. Dale, Aug 25 2024 *)

Formula

a(n) = (A023039(n) - 1)/2. - Max Alekseyev, Nov 13 2009

G.f.: -4*x*(x+1)/((x-1)*(x^2-18*x+1)). - Colin Barker, Oct 24 2012

From Amiram Eldar, Jan 11 2022: (Start)

a(n) = 5*Fibonacci(3*n)^2/4 - 1 if n is odd and 5*Fibonacci(3*n)^2/4 if n is even.

A000217(a(n)) = A292443(n). (End)

a(n) = (Lucas(6*n)-2)/4. - Jeffrey Shallit, Jan 20 2023

a(n) = 4 * A049683(n). - Alois P. Heinz, Jan 20 2023

Extensions

More terms from Max Alekseyev, Nov 13 2009

New definition by Antti Karttunen, Oct 24 2012