A144839 a(n) = Lucas(7^n).

29, 17393796001, 481682208844384447843365760878364816732549453120338354329505085763436029

Offset: 1

Author

Artur Jasinski, Sep 22 2008

Keywords

• nonn

Comments

Previous name was: a(n) = round(phi^(7^n)) where phi = 1.6180339887498948482... = (sqrt(5)+1)/2.

Links

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

Crossrefs

Cf. A000032, A001622, A001566, A006267, A144836, A144837, A144838, A144839.

Programs

Maple

a := proc(n) option remember; if n = 0 then 1 else a(n-1)^7 + 7*a(n-1)^5 + 14*a(n-1)^3 +7*a(n-1) end if; end;
seq(a(n), n = 1..5); # Peter Bala, Nov 28 2022

Mathematica

Table[Round[GoldenRatio^(7^n)], {n, 1, 5}]
c = (1 + Sqrt[5])/2; Table[Expand[c^(7^n) + (1 - c)^(7^n)], {n, 1, 5}] (* Artur Jasinski, Oct 05 2008 *)
Table[LucasL[7^n], {n, 1, 4}] (* Amiram Eldar, Jul 13 2025 *)

Formula

a(n) = G^(7^n) + (1 - G)^(7^n) = G^(7^n) + (-G)^(-7^n) where G is the golden ratio A001622. [Artur Jasinski, Oct 05 2008]

From Peter Bala, Nov 28 2022: (Start)

a(n) = Lucas(7^n).

a(n+1) = a(n)^7 + 7*a(n)^5 + 14*a(n)^3 + 7*a(n) with a(0) = 1.

a(n) == 1 (mod 7).

a(n+1) == a(n) (mod 7^(n+1)) for n >= 1 (a particular case of the Gauss congruences for the Lucas numbers).

In the ring of 7-adic integers, the limit_{n -> oo} a(n) exists and is a root of the quartic equation x^4 + 4*x^2 + 2 = 0. (End)

Extensions

New name from Peter Bala, Nov 28 2022