OFFSET
0,2
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).
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.
a(n) = (Lucas(6*n)-2)/4. - Jeffrey Shallit, Jan 20 2023
a(n) = 4 * A049683(n). - Alois P. Heinz, Jan 20 2023
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Mohamed Bouhamida, Nov 14 2007
EXTENSIONS
More terms from Max Alekseyev, Nov 13 2009
New definition by Antti Karttunen, Oct 24 2012
STATUS
approved