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A228489 Period of trace(n*tau), where tau = (1+sqrt(5))/2 = golden ratio.

%I A228489 #8 Nov 19 2023 21:15:55
%S A228489 1,1,1,1,1,4,1,1,2,4,1,1,1,6,3,1,2,1,1,10,1,8,6,1,9,4,10,8,1,18,1,6,6,
%T A228489 1,13,1,4,1,3,10,4,4,12,6,17,1,1,1,18,21,8,10,6,8,1,8,12,4,16,16,3,6,
%U A228489 1,14,9,14,20,1,6,36,18,1,4,13,26,1,12,20,20
%N A228489 Period of trace(n*tau), where tau = (1+sqrt(5))/2 = golden ratio.
%C A228489 It is assumed that trace(n*tau) is purely periodic, as conjectured at A228487 where trace is defined.
%e A228489 a(6) = 4 because the trace(6*tau) = 1001 (repeated) has period 4.
%e A228489    n  trace(n*tau)
%e A228489    -  ------------
%e A228489    1  000000000...
%e A228489    2  111111111...
%e A228489    3  000000000...
%e A228489    4  000000000...
%e A228489    5  000000000...
%e A228489    6  1001 (repeated)
%e A228489    7  000000000...
%e A228489    8  111111111...
%e A228489    9  10 (repeated)
%e A228489   10  0100 (repeated)
%e A228489   14  110011 (repeated)
%e A228489   17  10 (repeated)
%e A228489   20  0101010010 (repeated)
%e A228489   30  101000111111001010 (repeated)
%e A228489   31  000000000...
%e A228489   35  1100101010011 (repeated)
%t A228489 $MaxExtraPrecision = Infinity; period[seq_] := (If[Last[#1] == {} || Length[#1] == Length[seq] - 1, 0, Length[#1]] &)[NestWhileList[Rest, Rest[seq], #1 != Take[seq, Length[#1]] &, 1]]; periodicityReport[seq_] := ({Take[seq, Length[seq] - Length[#1]], period[#1], Take[#1, period[#1]]} &)[Take[seq, -Length[NestWhile[Rest[#1] &, seq, period[#1] == 0 &, 1, Length[seq]]]]]
%t A228489 (*output format {initial segment,period length,period}*)
%t A228489 t[{x_, y_, _}] := t[{x, y}]; t[{x_, y_}] := Prepend[If[# > y - #, {y - #, 1}, {#, 0}], y] &[Mod[x, y]]; userIn2[{x_, y_}] := Most[NestWhileList[t, {x, y}, (#[[2]] > 0) &]];
%t A228489 z = 160; pr = Table[p = Convergents[n*GoldenRatio, z];  pairs = Table[{Numerator[#], Denominator[#]} &[p[[k]]], {k, 1, z}]; periodicityReport[Most[Last[Map[Map[#[[3]] &, Rest[userIn2[#]]] &, pairs]]]], {n, 200}]
%t A228489 m = Map[#[[2]] &, pr]   (* _Peter J. C. Moses_, Aug 22 2013 *)
%Y A228489 Cf. A228487, A228488.
%K A228489 nonn
%O A228489 0,6
%A A228489 _Clark Kimberling_, Aug 23 2013