%I #12 Dec 04 2022 13:06:19

%S 1,1,3001,475400918060101145703001,

%T 29642179764875707696452732234250095350341524541114277856812964100763567848899514572925690068090872073476146381237687662210078001

%N a(n) = Fibonacci(5^n) / 5^n.

%C Numbers k such that k divides Fibonacci(k) are listed in A023172.

%C All powers of 5 belong to A023172.

%C 5^n divides Fibonacci(5^n).

%C a(n) == 1 (mod 1000).

%C {a(n+1)/a(n)} = {1, 3001, 158414167964045700001, 62351961552434956321060201440347372028390478647963811251289490034177804212636326088548682319305439375001, ...}.

%F a(n) = Fibonacci(5^n) / 5^n.

%F a(n+1) = 5^(4*n+1)*a(n)^5 - 5^(2*n+1)*a(n)^3 + a(n) with a(0) = 1. - _Peter Bala_, Nov 24 2022

%p a := proc(n) option remember; if n = 0 then 1 else 5^(4*n-3)*a(n-1)^5 - 5^(2*n-1)*a(n-1)^3 + a(n-1) end if; end proc: seq(a(n), n = 0..5); # _Peter Bala_, Nov 24 2022

%t Table[ Fibonacci[ 5^n ] / 5^n, {n,0,4} ]

%Y Cf. A023172, A121169, A121170, A058635, A045529.

%K nonn,easy

%O 0,3

%A _Alexander Adamchuk_, May 11 2007