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a(n) = F(F(n)) mod F(n), where F = Fibonacci = A000045.
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%I #19 Oct 29 2024 12:02:09

%S 0,0,1,2,0,5,12,5,33,5,1,0,232,233,55,5,1596,2563,1,5,987,10946,28656,

%T 0,0,75025,189653,89,1,6765,1,5,6765,1,9227460,0,24157816,1,63245985,

%U 5,1,267914275,433494436,4181,1134896405,1,2971215072,0,7778741816,75025

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

%H Alois P. Heinz, <a href="/A263101/b263101.txt">Table of n, a(n) for n = 1..2000</a>

%F a(n) = A007570(n) mod A000045(n).

%p F:= n-> (<<0|1>, <1|1>>^n)[1, 2]:

%p p:= (M, n, k)-> map(x-> x mod k, `if`(n=0, <<1|0>, <0|1>>,

%p `if`(n::even, p(M, n/2, k)^2, p(M, n-1, k).M))):

%p a:= n-> p(<<0|1>, <1|1>>, F(n)$2)[1, 2]:

%p seq(a(n), n=1..50);

%t F[n_] := MatrixPower[{{0, 1}, {1, 1}}, n][[1, 2]];

%t 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 a[n_] := p[{{0, 1}, {1, 1}}, F[n], F[n]][[1, 2]];

%t Table[a[n], {n, 1, 50}] (* _Jean-François Alcover_, Oct 29 2024, after _Alois P. Heinz_ *)

%Y Cf. A000045, A002708, A007570, A076240, A127787 (where a(n)=0), A263112, A274996.

%K nonn,look

%O 1,4

%A _Alois P. Heinz_, Oct 09 2015