A240836 Numbers n such that n^3 = x*y*z where 2 <= x <= y <= z , n^3+1 = (x-1)*(y+1)*(z+1).

2, 12, 80, 546, 3740, 25632, 175682, 1204140, 8253296, 56568930, 387729212, 2657535552, 18215019650, 124847601996, 855718194320, 5865179758242, 40200540113372, 275538601035360, 1888569667134146, 12944449068903660, 88722573815191472, 608113567637436642

Offset

1,1

Comments

Also, z/y approx = y/x approx = golden ratio.

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

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

Formula

a(n) = 2*F(2n)*F(2n-1) where F(n) are the Fibonacci numbers (A000045).

G.f.: 2*x*(1-2*x)/((1-x)*(1-7*x+x^2)). - Colin Barker, Apr 13 2014

a(n) = 2 * A081016(n-1). - Wesley Ivan Hurt, Apr 13 2014

Example

546^3 = 338 * 546 * 882, 546^3 + 1 = 337 * 547 * 883.
25632^3 = 15842 * 25632 * 41472, 25632^3 + 1 = 15841 * 25633 * 41473.

Maple

with(combinat); A240836:=n->2*fibonacci(2*n)*fibonacci(2*n-1); seq(A240836(n), n=1..30); # Wesley Ivan Hurt, Apr 13 2014

Mathematica

Table[2Fibonacci[2n]Fibonacci[2n-1], {n, 30}] (* Wesley Ivan Hurt, Apr 13 2014 *)

Prog

(PARI) vector(30, n, f=fibonacci; 2*f(2*n)*f(2*n-1)) \\ G. C. Greubel, Jul 15 2019
(Magma) F:=Fibonacci; [2*F(2*n)*F(2*n-1): n in [1..30]]; // G. C. Greubel, Jul 15 2019
(Sage) f=fibonacci; [2*f(2*n)*f(2*n-1) for n in (1..30)] # G. C. Greubel, Jul 15 2019
(GAP) F:=Fibonacci;; List([1..30], n-> 2*F(2*n)*F(2*n-1) ); # G. C. Greubel, Jul 15 2019

Crossrefs

Cf. A000045, A079472, A081016, A110035, A126116.

Sequence in context: A289662 A107632 A270775 * A270919 A082142 A069723

Adjacent sequences: A240833 A240834 A240835 * A240837 A240838 A240839

Keyword

nonn,easy

Author

Naohiro Nomoto, Apr 12 2014

Extensions

More terms from Colin Barker, Apr 13 2014

Status

approved