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A263112 a(n) = F(F(n)) mod n, where F = Fibonacci = A000045.

%I A263112 #19 Oct 29 2024 10:27:20
%S A263112 0,1,1,2,0,3,2,2,1,5,1,0,8,13,10,2,12,15,5,10,1,1,1,0,0,25,1,2,5,15,
%T A263112 27,2,10,33,20,0,1,1,34,10,40,21,18,2,10,1,1,0,1,25,1,2,16,21,5,26,37,
%U A263112 1,7,0,33,27,1,2,40,21,5,2,1,15,1,0,46,1,25,2,68
%N A263112 a(n) = F(F(n)) mod n, where F = Fibonacci = A000045.
%H A263112 Alois P. Heinz, <a href="/A263112/b263112.txt">Table of n, a(n) for n = 1..10000</a>
%F A263112 a(n) = A007570(n) mod n.
%p A263112 F:= n-> (<<0|1>, <1|1>>^n)[1, 2]:
%p A263112 p:= (M, n, k)-> map(x-> x mod k, `if`(n=0, <<1|0>, <0|1>>,
%p A263112           `if`(n::even, p(M, n/2, k)^2, p(M, n-1, k).M))):
%p A263112 a:= n-> p(<<0|1>, <1|1>>, F(n), n)[1, 2]:
%p A263112 seq(a(n), n=1..80);
%t A263112 F[n_] := MatrixPower[{{0, 1}, {1, 1}}, n][[1, 2]];
%t A263112 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 A263112 a[n_] := p[{{0, 1}, {1, 1}}, F[n], n][[1, 2]];
%t A263112 Table[a[n], {n, 1, 80}] (* _Jean-François Alcover_, Oct 29 2024, after _Alois P. Heinz_ *)
%Y A263112 Cf. A000045, A002708, A007570, A023172 (where a(n)=0), A263101, A274996, A338736.
%K A263112 nonn,look
%O A263112 1,4
%A A263112 _Alois P. Heinz_, Oct 09 2015