A292412 Numbers of the form Fibonacci(2*k-1) or Lucas(2*k-1); i.e., union of sequences A001519 and A002878.

1, 1, 2, 4, 5, 11, 13, 29, 34, 76, 89, 199, 233, 521, 610, 1364, 1597, 3571, 4181, 9349, 10946, 24476, 28657, 64079, 75025, 167761, 196418, 439204, 514229, 1149851, 1346269, 3010349, 3524578, 7881196, 9227465, 20633239, 24157817, 54018521, 63245986, 141422324

Offset: 1

Author

Vincenzo Librandi, Sep 16 2017

Keywords

• nonn
• easy

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) ).

Examples

			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.
		

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).

Crossrefs

Cf. A000045, A001519, A002878.

Programs

Magma

&cat[[Fibonacci(2*n-1), Lucas(2*n-1)]: n in [1..30]];

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 *)

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

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)