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A263101 a(n) = F(F(n)) mod F(n), where F = Fibonacci = A000045.

%I A263101 #23 Dec 13 2024 12:39:43
%S A263101 0,0,1,2,0,5,12,5,33,5,1,0,232,233,55,5,1596,2563,1,5,987,10946,28656,
%T A263101 0,0,75025,189653,89,1,6765,1,5,6765,1,9227460,0,24157816,1,63245985,
%U A263101 5,1,267914275,433494436,4181,1134896405,1,2971215072,0,7778741816,75025
%N A263101 a(n) = F(F(n)) mod F(n), where F = Fibonacci = A000045.
%H A263101 Alois P. Heinz, <a href="/A263101/b263101.txt">Table of n, a(n) for n = 1..2000</a>
%F A263101 a(n) = A007570(n) mod A000045(n).
%p A263101 F:= n-> (<<0|1>, <1|1>>^n)[1, 2]:
%p A263101 p:= (M, n, k)-> map(x-> x mod k, `if`(n=0, <<1|0>, <0|1>>,
%p A263101           `if`(n::even, p(M, n/2, k)^2, p(M, n-1, k).M))):
%p A263101 a:= n-> p(<<0|1>, <1|1>>, F(n)$2)[1, 2]:
%p A263101 seq(a(n), n=1..50);
%t A263101 F[n_] := MatrixPower[{{0, 1}, {1, 1}}, n][[1, 2]];
%t A263101 p[M_, n_, k_] := Mod[#, k]& /@ If[n == 0, {{1, 0}, {0, 1}}, If[EvenQ[n], MatrixPower[p[M, n/2, k], 2], p[M, n - 1, k].M]];
%t A263101 a[n_] := p[{{0, 1}, {1, 1}}, F[n], F[n]][[1, 2]];
%t A263101 Table[a[n], {n, 1, 50}] (* _Jean-François Alcover_, Oct 29 2024, after _Alois P. Heinz_ *)
%o A263101 (PARI) alist(nn)= my(f=fibonacci); [ f(f(n))%f(n) |n<-[1..nn] ]; \\ _Ruud H.G. van Tol_, Dec 13 2024
%Y A263101 Cf. A000045, A002708, A007570, A076240, A127787 (where a(n)=0), A263112, A274996.
%K A263101 nonn,look
%O A263101 1,4
%A A263101 _Alois P. Heinz_, Oct 09 2015