%I #48 Jun 09 2024 13:21:08
%S 1,5,31,209,1429,9791,67105,459941,3152479,21607409,148099381,
%T 1015088255,6957518401,47687540549,326855265439,2240299317521,
%U 15355239957205,105246380382911,721369422723169,4944339578679269,33889007628031711,232278713817542705
%N Numbers w such that (F(2n+1)^2, -F(2n)^2, -w) are primitive solutions of the Diophantine equation 2*x^3 + 2*y^3 + z^3 = 1, where F(n) is the n-th Fibonacci number (A000045).
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (8,-8,1).
%F a(n) = (2*F(2*n+1)^6 - 2*F(2*n)^6 - 1)^(1/3).
%F From _Colin Barker_, Oct 01 2020: (Start)
%F G.f.: (1 - 3*x - x^2) / ((1 - x)*(1 - 7*x + x^2)).
%F a(n) = 8*a(n-1) - 8*a(n-2) + a(n-3) for n>2.
%F (End)
%F a(n) = 2*A081018(n) + 1. - _Hugo Pfoertner_, Oct 01 2020
%F a(n) = A064170(n+2) + A033888(n). - _Flávio V. Fernandes_, Jan 10 2021
%F a(n) = F(2*n+1)*F(2*n+2) - F(2*n)^2. - _Wolfgang Berndt_, May 26 2023
%F a(2*n-1) = 5 + 6*Sum_{k=1..n-1} F(8*k+1), a(2*n) = 1 + 6*Sum_{k=1..n} F(8*k-3). - _XU Pingya_, Jun 09 2024
%e 2*(F(5)^2)^3 + 2*(-F(4)^2)^3 + (-31)^3 = 2*(25)^3 + 2*(-9)^3 + (-31)^3 = 1, a(2) = 31.
%t Table[(2*Fibonacci[2n+1]^6 - 2*Fibonacci[2n]^6 - 1)^(1/3), {n, 0, 21}]
%t Table[(Fibonacci[2n+1]*Fibonacci[2n+2]- Fibonacci[2n]^2), {n, 0, 21}] (* _Wolfgang Berndt_, May 26 2023 *)
%t LinearRecurrence[{8,-8,1},{1,5,31},30] (* _Harvey P. Dale_, Dec 17 2023 *)
%o (PARI) Vec((1 - 3*x - x^2) / ((1 - x)*(1 - 7*x + x^2)) + O(x^20)) \\ _Colin Barker_, Oct 01 2020
%Y Cf. A000045, A000578, A056573, A081018, A337929.
%K nonn,easy
%O 0,2
%A _XU Pingya_, Sep 30 2020