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A228489 Period length of trace(n*tau), where tau = (1+sqrt(5))/2 = golden ratio. 2
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, 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, 1, 14, 9, 14, 20, 1, 6, 36, 18, 1, 4, 13, 26, 1, 12, 20, 20 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

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

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

n .......... trace(n*tau)

1 .......... 000000000...

2 .......... 111111111...

3 .......... 000000000...

4 .......... 000000000...

5 .......... 000000000...

6 .......... 1001(repeated)

7 .......... 000000000...

8 .......... 111111111...

9 .......... 10(repeated)

10 ......... 0100(repeated)

14 ......... 110011(repeated)

17 ......... 10(repeated)

20 ......... 0101010010(repeated)

30 ......... 101000111111001010(repeated)

31 ......... 000000000...

35 ......... 1100101010011(repeated)

LINKS

Table of n, a(n) for n=0..78.

MATHEMATICA

$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]]]]]

(*output format {initial segment, period length, period}*)

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) &]];

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}]

m = Map[#[[2]] &, pr]   (* Peter J. C. Moses, Aug 22 2013 *)

CROSSREFS

Cf. A228487, A228488.

Sequence in context: A222171 A325529 A264534 * A096103 A204456 A143441

Adjacent sequences:  A228486 A228487 A228488 * A228490 A228491 A228492

KEYWORD

nonn

AUTHOR

Clark Kimberling, Aug 23 2013

STATUS

approved

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Last modified August 14 07:59 EDT 2020. Contains 336477 sequences. (Running on oeis4.)