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A354336 a(n) is the integer w such that (L(2*n)^2, -L(2*n-1)^2, -w) is a primitive solution to the Diophantine equation 2*x^3 + 2*y^3 + z^3 = 125, where L(n) is the n-th Lucas number (A000032).

%I A354336 #27 Mar 22 2025 19:03:56
%S A354336 1,11,61,401,2731,18701,128161,878411,6020701,41266481,282844651,
%T A354336 1938646061,13287677761,91075098251,624238009981,4278590971601,
%U A354336 29325898791211,201002700566861,1377693005176801,9442848335670731,64722245344518301,443612869075957361
%N A354336 a(n) is the integer w such that (L(2*n)^2, -L(2*n-1)^2, -w) is a primitive solution to the Diophantine equation 2*x^3 + 2*y^3 + z^3 = 125, where L(n) is the n-th Lucas number (A000032).
%C A354336 Subsequence of A017281.
%H A354336 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (8,-8,1).
%F A354336 a(n) = (-125 + 2*A005248(n)^6 - 2*A002878(n-1)^6)^(1/3).
%F A354336 a(n) = Lucas(4*n+1) - Lucas(4*n-2) + 3 = A056914(n) - 15*A092521(n-1), for n > 1.
%F A354336 a(n) = Lucas(4*n+1) + 1 - Lucas(2*n-1)^2.
%F A354336 a(n) = 2*A081015(n-1) + 1.
%F A354336 a(n) = 8*a(n-1) - 8*a(n-2) + a(n-3).
%F A354336 G.f.: (1 + 3*x - 19*x^2)/((1 - x)*(1 - 7*x + x^2)). - _Stefano Spezia_, Jun 22 2022
%F A354336 a(n) = (F(2*n+1) + F(2*n-1))^2 + (F(2*n+1) + F(2*n-1)) * (F(2*n-1) + F(2*n-3)) - (F(2*n-1) + F(2*n-3))^2. - _XU Pingya_, Jul 17 2024
%e A354336 2*(L(4)^2)^3 + 2*(-L(3)^2)^3 + (-61)^3 = 2*(49)^3 + 2*(-1)^3 + (-61)^3 = 125, a(2) = 61.
%t A354336 LucasL[4*Range[22]-3] + 1 - LucasL[2*Range[22]-3]^2
%Y A354336 Cf. A000032, A002878, A005248, A017281, A056914, A081015, A092521.
%Y A354336 Cf. A337928, A354337.
%K A354336 nonn,easy
%O A354336 0,2
%A A354336 _XU Pingya_, Jun 20 2022