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

0,2

Comments

Subsequence of A017281.

Links

Table of n, a(n) for n=0..21.

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

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

Example

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

Mathematica

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

Crossrefs

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

Cf. A337928, A354337.

Sequence in context: A156095 A068846 A092164 * A088545 A259257 A289646

Adjacent sequences: A354333 A354334 A354335 * A354337 A354338 A354339

Keyword

nonn,easy

Author

XU Pingya, Jun 20 2022

Status

approved