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

1,3

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

Cf. A000045, A001519, A002878.

Sequence in context: A287269 A136988 A074725 * A113733 A294434 A191289

Adjacent sequences: A292409 A292410 A292411 * A292413 A292414 A292415

Keyword

nonn,easy

Author

Vincenzo Librandi, Sep 16 2017

Status

approved