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A337928 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).

%I A337928 #48 Jun 09 2024 13:21:08
%S A337928 1,5,31,209,1429,9791,67105,459941,3152479,21607409,148099381,
%T A337928 1015088255,6957518401,47687540549,326855265439,2240299317521,
%U A337928 15355239957205,105246380382911,721369422723169,4944339578679269,33889007628031711,232278713817542705
%N A337928 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 A337928 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (8,-8,1).
%F A337928 a(n) = (2*F(2*n+1)^6 - 2*F(2*n)^6 - 1)^(1/3).
%F A337928 From _Colin Barker_, Oct 01 2020: (Start)
%F A337928 G.f.: (1 - 3*x - x^2) / ((1 - x)*(1 - 7*x + x^2)).
%F A337928 a(n) = 8*a(n-1) - 8*a(n-2) + a(n-3) for n>2.
%F A337928 (End)
%F A337928 a(n) = 2*A081018(n) + 1. - _Hugo Pfoertner_, Oct 01 2020
%F A337928 a(n) = A064170(n+2) + A033888(n). - _Flávio V. Fernandes_, Jan 10 2021
%F A337928 a(n) = F(2*n+1)*F(2*n+2) - F(2*n)^2. - _Wolfgang Berndt_, May 26 2023
%F A337928 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 A337928 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 A337928 Table[(2*Fibonacci[2n+1]^6 - 2*Fibonacci[2n]^6 - 1)^(1/3), {n, 0, 21}]
%t A337928 Table[(Fibonacci[2n+1]*Fibonacci[2n+2]- Fibonacci[2n]^2), {n, 0, 21}] (* _Wolfgang Berndt_, May 26 2023 *)
%t A337928 LinearRecurrence[{8,-8,1},{1,5,31},30] (* _Harvey P. Dale_, Dec 17 2023 *)
%o A337928 (PARI) Vec((1 - 3*x - x^2) / ((1 - x)*(1 - 7*x + x^2)) + O(x^20)) \\ _Colin Barker_, Oct 01 2020
%Y A337928 Cf. A000045, A000578, A056573, A081018, A337929.
%K A337928 nonn,easy
%O A337928 0,2
%A A337928 _XU Pingya_, Sep 30 2020