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

%I

%S 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 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 1,14,9,14,20,1,6,36,18,1,4,13,26,1,12,20,20

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

%C It is assumed that trace(n*tau) is purely periodic, as conjectured at A228487 where trace is defined.

%C a(6) = 4 because the trace(6*tau) = 1001(repeated) has period length 4. Examples:

%C n .......... trace(n*tau)

%C 1 .......... 000000000...

%C 2 .......... 111111111...

%C 3 .......... 000000000...

%C 4 .......... 000000000...

%C 5 .......... 000000000...

%C 6 .......... 1001(repeated)

%C 7 .......... 000000000...

%C 8 .......... 111111111...

%C 9 .......... 10(repeated)

%C 10 ......... 0100(repeated)

%C 14 ......... 110011(repeated)

%C 17 ......... 10(repeated)

%C 20 ......... 0101010010(repeated)

%C 30 ......... 101000111111001010(repeated)

%C 31 ......... 000000000...

%C 35 ......... 1100101010011(repeated)

%t \$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 (*output format {initial segment,period length,period}*)

%t 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 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 m = Map[#[[2]] &, pr] (* _Peter J. C. Moses_, Aug 22 2013 *)

%Y Cf. A228487, A228488.

%K nonn

%O 0,6

%A _Clark Kimberling_, Aug 23 2013

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Last modified September 25 15:31 EDT 2020. Contains 337344 sequences. (Running on oeis4.)