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

%I A240836 #26 Sep 08 2022 08:46:07
%S A240836 2,12,80,546,3740,25632,175682,1204140,8253296,56568930,387729212,
%T A240836 2657535552,18215019650,124847601996,855718194320,5865179758242,
%U A240836 40200540113372,275538601035360,1888569667134146,12944449068903660,88722573815191472,608113567637436642
%N A240836 Numbers n such that n^3 = x*y*z where 2 <= x <= y <= z , n^3+1 = (x-1)*(y+1)*(z+1).
%C A240836 Also, z/y approx = y/x approx = golden ratio.
%H A240836 G. C. Greubel, <a href="/A240836/b240836.txt">Table of n, a(n) for n = 1..1000</a>
%H A240836 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (8,-8,1).
%F A240836 a(n) = 2*F(2n)*F(2n-1) where F(n) are the Fibonacci numbers (A000045).
%F A240836 G.f.: 2*x*(1-2*x)/((1-x)*(1-7*x+x^2)). - _Colin Barker_, Apr 13 2014
%F A240836 a(n) = 2 * A081016(n-1). - _Wesley Ivan Hurt_, Apr 13 2014
%e A240836 546^3 = 338 * 546 * 882, 546^3 + 1 = 337 * 547 * 883.
%e A240836 25632^3 = 15842 * 25632 * 41472, 25632^3 + 1 = 15841 * 25633 * 41473.
%p A240836 with(combinat); A240836:=n->2*fibonacci(2*n)*fibonacci(2*n-1); seq(A240836(n), n=1..30); # _Wesley Ivan Hurt_, Apr 13 2014
%t A240836 Table[2Fibonacci[2n]Fibonacci[2n-1], {n, 30}] (* _Wesley Ivan Hurt_, Apr 13 2014 *)
%o A240836 (PARI) vector(30, n, f=fibonacci; 2*f(2*n)*f(2*n-1)) \\ _G. C. Greubel_, Jul 15 2019
%o A240836 (Magma) F:=Fibonacci; [2*F(2*n)*F(2*n-1): n in [1..30]]; // _G. C. Greubel_, Jul 15 2019
%o A240836 (Sage) f=fibonacci; [2*f(2*n)*f(2*n-1) for n in (1..30)] # _G. C. Greubel_, Jul 15 2019
%o A240836 (GAP) F:=Fibonacci;; List([1..30], n-> 2*F(2*n)*F(2*n-1) ); # _G. C. Greubel_, Jul 15 2019
%Y A240836 Cf. A000045, A079472, A081016, A110035, A126116.
%K A240836 nonn,easy
%O A240836 1,1
%A A240836 _Naohiro Nomoto_, Apr 12 2014
%E A240836 More terms from _Colin Barker_, Apr 13 2014