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).

1, 5, 31, 209, 1429, 9791, 67105, 459941, 3152479, 21607409, 148099381, 1015088255, 6957518401, 47687540549, 326855265439, 2240299317521, 15355239957205, 105246380382911, 721369422723169, 4944339578679269, 33889007628031711, 232278713817542705

Offset

0,2

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

From Colin Barker, Oct 01 2020: (Start)

G.f.: (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>2.

(End)

a(n) = 2*A081018(n) + 1. - Hugo Pfoertner, Oct 01 2020

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

a(n) = F(2*n+1)*F(2*n+2) - F(2*n)^2. - Wolfgang Berndt, May 26 2023

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

Example

2*(F(5)^2)^3 + 2*(-F(4)^2)^3 + (-31)^3 = 2*(25)^3 + 2*(-9)^3 + (-31)^3 = 1, a(2) = 31.

Mathematica

Table[(2*Fibonacci[2n+1]^6 - 2*Fibonacci[2n]^6 - 1)^(1/3), {n, 0, 21}]
Table[(Fibonacci[2n+1]*Fibonacci[2n+2]- Fibonacci[2n]^2), {n, 0, 21}] (* Wolfgang Berndt, May 26 2023 *)
LinearRecurrence[{8, -8, 1}, {1, 5, 31}, 30] (* Harvey P. Dale, Dec 17 2023 *)

Prog

(PARI) Vec((1 - 3*x - x^2) / ((1 - x)*(1 - 7*x + x^2)) + O(x^20)) \\ Colin Barker, Oct 01 2020

Crossrefs

Cf. A000045, A000578, A056573, A081018, A337929.

Sequence in context: A301420 A359713 A092636 * A178792 A007197 A247639

Adjacent sequences: A337925 A337926 A337927 * A337929 A337930 A337931

Keyword

nonn,easy

Author

XU Pingya, Sep 30 2020

Status

approved