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

%I A144839 #22 Jul 13 2025 11:08:36
%S A144839 29,17393796001,
%T A144839 481682208844384447843365760878364816732549453120338354329505085763436029
%N A144839 a(n) = Lucas(7^n).
%C A144839 Previous name was: a(n) = round(phi^(7^n)) where phi = 1.6180339887498948482... = (sqrt(5)+1)/2.
%H A144839 Amiram Eldar, <a href="/A144839/b144839.txt">Table of n, a(n) for n = 1..4</a>
%F A144839 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]
%F A144839 From _Peter Bala_, Nov 28 2022: (Start)
%F A144839 a(n) = Lucas(7^n).
%F A144839 a(n+1) = a(n)^7 + 7*a(n)^5 + 14*a(n)^3 + 7*a(n) with a(0) = 1.
%F A144839 a(n) == 1 (mod 7).
%F A144839 a(n+1) == a(n) (mod 7^(n+1)) for n >= 1 (a particular case of the Gauss congruences for the Lucas numbers).
%F A144839 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)
%p A144839 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;
%p A144839 seq(a(n), n = 1..5); # _Peter Bala_, Nov 28 2022
%t A144839 Table[Round[GoldenRatio^(7^n)], {n, 1, 5}]
%t A144839 c = (1 + Sqrt[5])/2; Table[Expand[c^(7^n) + (1 - c)^(7^n)], {n, 1, 5}] (* _Artur Jasinski_, Oct 05 2008 *)
%t A144839 Table[LucasL[7^n], {n, 1, 4}] (* _Amiram Eldar_, Jul 13 2025 *)
%Y A144839 Cf. A000032, A001622, A001566, A006267, A144836, A144837, A144838, A144839.
%K A144839 nonn
%O A144839 1,1
%A A144839 _Artur Jasinski_, Sep 22 2008
%E A144839 New name from _Peter Bala_, Nov 28 2022