|
|
A132584
|
|
a(0)=0, a(1)=4; for n > 1, a(n) = 18*a(n-1) - a(n-2) + 8.
|
|
5
|
|
|
0, 4, 80, 1444, 25920, 465124, 8346320, 149768644, 2687489280, 48225038404, 865363202000, 15528312597604, 278644263554880, 5000068431390244, 89722587501469520, 1610006506595061124, 28890394531209630720, 518417095055178291844, 9302617316461999622480
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
|
|
|
FORMULA
|
G.f.: -4*x*(x+1)/((x-1)*(x^2-18*x+1)). - Colin Barker, Oct 24 2012
a(n) = 5*Fibonacci(3*n)^2/4 - 1 if n is odd and 5*Fibonacci(3*n)^2/4 if n is even.
|
|
MATHEMATICA
|
|
|
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
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|