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

Author

Clark Kimberling, Aug 23 2013

Keywords

• nonn

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)

Examples

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

Crossrefs

Cf. A228487, A228489.

Programs

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 *)