OFFSET
1,3
COMMENTS
From the abstract of the Perrine reference: The Diophantine equation x^2 = 5*y^2 - 4 and its three classes of solutions for automorphs will be discussed. For n an odd positive integer, any ordered pair (x, y) = ( L(2*n-1), F(2*n-1) ) is a solution to the equation and all of the solutions are ( +-L(2*n-1), +-F(2*n-1) ).
LINKS
Colin Barker, Table of n, a(n) for n = 1..1000
Serge Perrine, Some properties of the equation x^2=5y^2-4, The Fibonacci Quarterly, Vol. 54, No. 2 (2016) pp. 172-177.
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-1).
FORMULA
From Colin Barker, Sep 16 2017: (Start)
G.f.: x*(1 + x - x^2 + x^3)/((1 + x - x^2)*(1 - x - x^2)).
a(n) = 3*a(n-2) - a(n-4) for n>4. (End)
EXAMPLE
2 and 4 are in sequence because 5*2^2 - 4 = 4^2.
5 and 11 are in sequence because 5*5^2 - 4 = 11^2.
MATHEMATICA
Join[{1}, z=50; s=Table[LucasL[2 h - 1], {h, 1, z}]; t=Table[Fibonacci[2 k - 1], {k, 1, z}]; v=Union[t, s]]
{Fibonacci[#], LucasL[#]}&/@(2*Range[20]-1)//Flatten (* Harvey P. Dale, Jul 18 2020 *)
PROG
(Magma) &cat[[Fibonacci(2*n-1), Lucas(2*n-1)]: n in [1..30]];
(PARI) Vec(x*(1 + x - x^2 + x^3) / ((1 + x - x^2)*(1 - x - x^2)) + O(x^100)) \\ Colin Barker, Sep 18 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Vincenzo Librandi, Sep 16 2017
STATUS
approved