A228488 Period length of trace(sqrt(n)).

0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 3, 1, 1, 0, 1, 1, 4, 1, 2, 4, 1, 1, 0, 1, 1, 1, 4, 1, 6, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 6, 1, 4, 8, 1, 1, 0, 1, 1, 4, 4, 4, 1, 1, 2, 4, 4, 1, 7, 1, 1, 0, 1, 1, 8, 1, 6, 4, 6, 1, 5, 3, 1, 8, 4, 1, 1, 1, 0, 1, 1, 1, 4, 6

Offset

0,13

Comments

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

If n is a square, then trace(sqrt(n)) is the empty word, denoted by E. Examples:

n ........... trace(sqrt(n))

1 ........... E

2 ........... 000000000...

3 ........... 111111111...

4 ........... E

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

6 ........... 000000000...

7 ........... 111111111...

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

9 ........... E

10 .......... 000000000...

11 .......... 000000000...

12 .......... 000000000...

13 .......... 110(repeated)

19 .......... 0110(repeated)

22 .......... 1001(repeated)

31 .......... 100010(repeated)

46 .......... 11000101(repeated)

Links

Table of n, a(n) for n=0..85.

Example

a(13) = 3 because the trace(sqrt(13)) = 110(repeated) has period length 3.

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[If[IntegerQ[Sqrt[n]], {0, 0}, p = Convergents[Sqrt[n], z]; pairs = Table[{Numerator[#], Denominator[#]} &[p[[k]]], {k, 1, z}]; periodicityReport[    Most[Last[Map[Map[#[[3]] &, Rest[userIn2[#]]] &, pairs]]]]], {n, 120}]
m = Map[#[[2]] &, pr]  (* Peter J. C. Moses, Aug 22 2013 *)

Crossrefs

Cf. A228487, A228489.

Sequence in context: A319993 A122776 A261609 * A338638 A062172 A196838

Adjacent sequences: A228485 A228486 A228487 * A228489 A228490 A228491

Keyword

nonn

Author

Clark Kimberling, Aug 23 2013

Status

approved