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 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 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 A284376 Adjacent sequences:  A228485 A228486 A228487 * A228489 A228490 A228491 KEYWORD nonn AUTHOR Clark Kimberling, Aug 23 2013 STATUS approved

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