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A292412 Numbers of the form Fibonacci(2*k-1) or Lucas(2*k-1); i.e., union of sequences A001519 and A002878. 1
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 (list; graph; refs; listen; history; text; internal format)
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) ).

REFERENCES

Serge Perrine, Some properties of the equation x^2 = 5*y^2 - 4, Fibonacci Quart. 54 (2016), no. 2, 172-177.

LINKS

Colin Barker, Table of n, a(n) for n = 1..1000

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]]

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

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Last modified September 17 17:32 EDT 2019. Contains 327136 sequences. (Running on oeis4.)