A144838 a(n) = Lucas(6^n).

18, 33385282, 1384619022984618483717737087933569992335566082

Offset: 1

Author

Artur Jasinski, Sep 22 2008

Keywords

• nonn

Comments

Previous name was: a(n) = round(phi^(6^n)) where phi = 1.6180339887498948482... = (sqrt(5)+1)/2.

General (hyperbolic) trigonometric formula for a(n) = round(phi^((2*k)^n)) = 2*cosh((2*k)^n*arccosh(sqrt(5)/2)) where phi = 1.6180339887498948482... = (sqrt(5)+1)/2. - Artur Jasinski, Oct 09 2008

Links

• Amiram Eldar, Table of n, a(n) for n = 1..4

Crossrefs

Cf. A001566, A006267, A144836, A144837, A144838, A144839.

Programs

Maple

a := proc(n) option remember; if n = 1 then 18 else a(n-1)^6 - 6*a(n-1)^4 + 9*a(n-1)^2 - 2 end if; end;
seq(a(n), n = 1..5); # Peter Bala, Nov 28 2022

Mathematica

Table[Round[GoldenRatio^(6^n)], {n, 1, 5}]
c = (1 + Sqrt[5])/2; Table[Expand[c^(6^n) + (1 - c)^(6^n)], {n, 1, 5}] (* Artur Jasinski, Oct 05 2008 *)
Table[Round[2*Cosh[6^n*ArcCosh[Sqrt[5]/2]]], {n, 1, 4}] (* Artur Jasinski, Oct 09 2008 *)
Table[LucasL[6^n], {n, 1, 4}] (* Amiram Eldar, Jul 13 2025 *)

Formula

a(n) = G^(6^n) + (1 - G)^(6^n) = G^(6^n) + (-G)^(-6^n) where G is the golden ratio A001622. - Artur Jasinski, Oct 05 2008

a(n) = 2*cosh(6^n*arccosh(sqrt(5)/2)). - Artur Jasinski, Oct 09 2008

From Peter Bala, Nov 28 2022: (Start)

a(n) = Lucas(6^n).

a(n+1) = a(n)^6 - 6*a(n)^4 + 9*a(n)^2 - 2 with a(1) = 18. (End)

Extensions

New name from Peter Bala, Nov 28 2022