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A132634 a(n) = Fibonacci(n) mod n^2.

%I A132634 #27 Oct 27 2023 21:49:36
%S A132634 0,1,2,3,5,8,13,21,34,55,89,0,64,181,160,219,152,316,210,365,362,287,
%T A132634 91,288,25,389,317,291,378,440,869,261,574,339,765,432,443,533,1285,
%U A132634 1355,1641,1504,85,1741,20,551,1832,576,1457,1525,389,803,2066,332,1820,245
%N A132634 a(n) = Fibonacci(n) mod n^2.
%C A132634 a(n)=0 for n=1 and n=12 only (conjecture).
%H A132634 Alois P. Heinz, <a href="/A132634/b132634.txt">Table of n, a(n) for n = 1..20000</a> (first 1000 terms from Hieronymus Fischer)
%e A132634 a(13) = 64, since Fibonacci(13) = 233 == 64 (mod 13^2).
%p A132634 p:= (M, n, k)-> map(x-> x mod k, `if`(n=0, <<1|0>, <0|1>>,
%p A132634           `if`(n::even, p(M, n/2, k)^2, p(M, n-1, k).M))):
%p A132634 a:= n-> p(<<0|1>, <1|1>>, n, n^2)[1, 2]:
%p A132634 seq(a(n), n=1..80);
%t A132634 Table[Mod[Fibonacci[n],n^2],{n,200}] (* _Vladimir Joseph Stephan Orlovsky_, Nov 28 2010 *)
%Y A132634 Cf. A000045, A023173, A236395.
%K A132634 nonn
%O A132634 1,3
%A A132634 _Hieronymus Fischer_, Aug 24 2007