

A183110


Periodlength of the ultimate periodic behavior of the orbit of a list [1,1,1,...,1] of n 1's under the mapping defined in the comments.


3



1, 2, 1, 3, 3, 1, 4, 4, 4, 1, 5, 5, 5, 5, 1, 6, 6, 6, 6, 6, 1, 7, 7, 7, 7, 7, 7, 1, 8, 8, 8, 8, 8, 8, 8, 1, 9, 9, 9, 9, 9, 9, 9, 9, 1, 10, 10, 10, 10, 10, 10, 10, 10, 10, 1, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 1, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 1, 13, 13
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OFFSET

1,2


COMMENTS

We use the list mapping introduced in A092964, whereby one removes the first term of the list, z(1), and adds 1 to each of the next z(1) terms (appending 1's if necessary) to get a new list.
This is also conjectured to be the length of the longest cycle of pebblemoves among the partitions of n (cf. A201144).  Andrew V. Sutherland


LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..1000


FORMULA

It appears, but has not yet been proved, that a(n)=1 if n=t(k) and a(n)=k if t(k1) < n < t(k) where t(k) is the kth triangular number t(k) = k*(k+1)/2.


EXAMPLE

Under the indicated mapping the list [1,1,1,1,1,1,1] of seven 1's results in the orbit [1,1,1,1,1,1,1], [2,1,1,1,1,1], [2,2,1,1,1], [3,2,1,1], [3,2,2], [3,3,1], [4,2,1], [3,2,1,1], ... which is clearly periodic with periodlength 4, so a(7) = 4.


MATHEMATICA

f[p_] := Module[{pp, x, lpp, m, i}, pp = p; x = pp[[1]]; pp = Delete[pp, 1]; lpp = Length[pp]; m = Min[x, lpp]; For[i = 1, i ≤ m, i++, pp[[i]]++]; For[i = 1, i ≤ x  lpp, i++, AppendTo[pp, 1]]; pp]; orb[p_] := Module[{s, v}, v = p; s = {v}; While[! MemberQ[s, v = f[v]], AppendTo[s, v]]; s]; attractor[p_] := Module[{orbp, pos, len, per}, orbp = orb[p]; pos = Flatten[Position[orbp, f[orbp[[1]]]]][[1]]  1; (*pos = steps to enter period*) len = Length[orbp]  pos; per = Take[orbp, len]; Sort[per]]; a = {}; For[n = 1, n ≤ 80, n++, {rn = Table[1, {k, 1, n}]; orbn = orb[rn]; lenorb = Length[orbn]; lenattr = Length[attractor[rn]]; AppendTo[a, lenattr]}]; Print[a];


CROSSREFS

Cf. A037306, A092964, A178572, A178574.
Sequence in context: A181108 A211782 A211701 * A117895 A188002 A186974
Adjacent sequences: A183107 A183108 A183109 * A183111 A183112 A183113


KEYWORD

nonn,easy


AUTHOR

John W. Layman, Feb 01 2011


STATUS

approved



