A337929 Numbers w such that (F(2*n-1)^2, -F(2*n)^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).

1, 11, 79, 545, 3739, 25631, 175681, 1204139, 8253295, 56568929, 387729211, 2657535551, 18215019649, 124847601995, 855718194319, 5865179758241, 40200540113371, 275538601035359, 1888569667134145, 12944449068903659, 88722573815191471, 608113567637436641

Offset

1,2

Links

Table of n, a(n) for n=1..22.

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

Formula

a(n) = (2*F(2*n)^6 - 2*F(2*n-1)^6 + 1)^(1/3).

From Colin Barker, Oct 01 2020: (Start)

G.f.: x*(1 + 3*x - x^2) / ((1 - x)*(1 - 7*x + x^2)).

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

(End)

a(n) = 2*A003482(n) + 1. - Hugo Pfoertner, Oct 01 2020

a(n) = A033888(n) - A064170(n+2). - Flávio V. Fernandes, Jan 10 2021

Example

2*(F(3)^2)^3 + 2*(-F(4)^2)^3 + 11^3 = 2*4^3 + 2*(-9)^3 + 11^3 = 1, 11 is a term.

Mathematica

Table[(2*Fibonacci[2n]^6 - 2*Fibonacci[2n-1]^6 + 1)^(1/3), {n, 22}]
LinearRecurrence[{8, -8, 1}, {1, 11, 79}, 30] (* Harvey P. Dale, Aug 23 2021 *)

Crossrefs

Cf. A000045, A000578, A003482, A056573, A337928.

Sequence in context: A243416 A101028 A125348 * A155619 A126506 A159663

Adjacent sequences: A337926 A337927 A337928 * A337930 A337931 A337932

Keyword

nonn,easy

Author

XU Pingya, Sep 30 2020

Status

approved