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).

1, 11, 61, 401, 2731, 18701, 128161, 878411, 6020701, 41266481, 282844651, 1938646061, 13287677761, 91075098251, 624238009981, 4278590971601, 29325898791211, 201002700566861, 1377693005176801, 9442848335670731, 64722245344518301, 443612869075957361

Offset: 0

Author

XU Pingya, Jun 20 2022

Keywords

• nonn
• easy

Comments

Subsequence of A017281.

Examples

			2*(L(4)^2)^3 + 2*(-L(3)^2)^3 + (-61)^3 = 2*(49)^3 + 2*(-1)^3 + (-61)^3 = 125, a(2) = 61.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (8,-8,1).

Crossrefs

Cf. A000032, A002878, A005248, A017281, A056914, A081015, A092521.

Cf. A337928, A354337.

Programs

Mathematica

LucasL[4*Range[22]-3] + 1 - LucasL[2*Range[22]-3]^2

Formula

a(n) = (-125 + 2*A005248(n)^6 - 2*A002878(n-1)^6)^(1/3).

a(n) = Lucas(4*n+1) - Lucas(4*n-2) + 3 = A056914(n) - 15*A092521(n-1), for n > 1.

a(n) = Lucas(4*n+1) + 1 - Lucas(2*n-1)^2.

a(n) = 2*A081015(n-1) + 1.

a(n) = 8*a(n-1) - 8*a(n-2) + a(n-3).

G.f.: (1 + 3*x - 19*x^2)/((1 - x)*(1 - 7*x + x^2)). - Stefano Spezia, Jun 22 2022

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