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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.

%I A292412 #34 Jan 05 2025 19:51:41
%S A292412 1,1,2,4,5,11,13,29,34,76,89,199,233,521,610,1364,1597,3571,4181,9349,
%T A292412 10946,24476,28657,64079,75025,167761,196418,439204,514229,1149851,
%U A292412 1346269,3010349,3524578,7881196,9227465,20633239,24157817,54018521,63245986,141422324
%N A292412 Numbers of the form Fibonacci(2*k-1) or Lucas(2*k-1); i.e., union of sequences A001519 and A002878.
%C A292412 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) ).
%H A292412 Colin Barker, <a href="/A292412/b292412.txt">Table of n, a(n) for n = 1..1000</a>
%H A292412 Serge Perrine, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/54-2/Perrine02242016.pdf">Some properties of the equation x^2=5y^2-4</a>, The Fibonacci Quarterly, Vol. 54, No. 2 (2016) pp. 172-177.
%H A292412 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,0,-1).
%F A292412 From _Colin Barker_, Sep 16 2017: (Start)
%F A292412 G.f.: x*(1 + x - x^2 + x^3)/((1 + x - x^2)*(1 - x - x^2)).
%F A292412 a(n) = 3*a(n-2) - a(n-4) for n>4. (End)
%e A292412 2 and 4 are in sequence because 5*2^2 - 4 = 4^2.
%e A292412 5 and 11 are in sequence because 5*5^2 - 4 = 11^2.
%t A292412 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]]
%t A292412 {Fibonacci[#],LucasL[#]}&/@(2*Range[20]-1)//Flatten (* _Harvey P. Dale_, Jul 18 2020 *)
%o A292412 (Magma) &cat[[Fibonacci(2*n-1), Lucas(2*n-1)]: n in [1..30]];
%o A292412 (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
%Y A292412 Cf. A000045, A001519, A002878.
%K A292412 nonn,easy
%O A292412 1,3
%A A292412 _Vincenzo Librandi_, Sep 16 2017