login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A228488 Period length of trace(sqrt(n)). 4
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 (list; graph; refs; listen; history; text; internal format)
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: A193140 A122776 A261609 * A062172 A196838 A088205

Adjacent sequences:  A228485 A228486 A228487 * A228489 A228490 A228491

KEYWORD

nonn

AUTHOR

Clark Kimberling, Aug 23 2013

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified March 29 15:10 EDT 2017. Contains 284273 sequences.