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%I #23 Jun 07 2019 19:36:07
%S 1,2,2,4,2,6,6,8,2,2,11,2,13,14,6,6,2,11,19,2,21,6,2,2,2,26,27,6,11,
%T 26,13,11,19,19,11,2,26,26,13,40,26,2,2,13,45,2,26,19,49,50,51,51,21,
%U 13,26,2,57,26,6,13,2,27,63,57,26,6,21,26,21,11,26,40,73,74,45,11,77,78,2,80,6,49,2,2,85,73,87,27,89
%N Self referencing version of the "Kimberling shuffle" sequence (see Comments).
%C If the first row of the expulsion array is replaced by this sequence, and the rows are "shuffled" then the sequence reappears in the diagonal.
%C For integer n >= 1 define the set [n]={x; A^r(x)=n}U{y; B^r(y)=n}; (r=0,1,2,3..., A^0(n)=B^0(n)=n), where A=A007063 and B=A006852 (mutual inverses). This set includes n, together with all numbers linked to n by A and B. If a number m is in [n], then [m]=[n], therefore we name the set by its least element k, which takes the following values: 1,2,4,6,8,11,13,14,19,21,26,27,40,45,48,50,51,57,63,... Assuming every n is a term in A, the collection of distinct sets [k] is a partition of the natural numbers, and this sequence is constructed by replacing in the first row of the original array, every number n in [k], with k.
%C A lexicographically earliest version can be obtained from this sequence by replacing any term > all preceding terms by k+1, where k is the greatest term seen so far. Thus: 1,2,2,3,2,4,4,5,2,2,6,2,7,8,4,4,2,6,9,2,10,4,2,2,2,11,...
%C From _Lars Blomberg_, Apr 27 2019: (Start)
%C Starting with some k value and extending in both directions using A and B results in a "valley" with k at the bottom and often sub-valleys on the hillsides (larger than k). (See the document referenced in A038807 for an illustration.)
%C So the k sequence is computed by selecting the smallest value not yet seen and iterate as far as possible, then select the next value not seen, etc.
%C However, while it seems that A and B values goes toward infinity, it is not known whether a known valley will eventually connect to another known valley, leading to a different set of k values.
%C The DATA is based on iterating A and B until the value > 10^8. (End)
%D R. K. Guy, Unsolved Problems Number Theory, Sect E35.
%H D. Gale, <a href="http://dx.doi.org/10.1007/978-1-4612-2192-0">Tracking the Automatic Ant: And Other Mathematical Explorations</a>, ch. 5, p. 27. Springer, 1998.
%H C. Kimberling, <a href="https://cms.math.ca/crux/backfile/Crux_v18n03_Mar.pdf">Problem 1615</a>, Crux Mathematicorum, Vol. 17 (2) 44 1991 and Vol. 18, March 1992, pp. 82-83.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/KimberlingSequence.html">Kimberling Sequence</a>
%e Examples of [k] for the above list up to k=27:
%e [1]={1}; so a(1)=1
%e [2]={2,3,5,9,10,12,17,20,23,24,25,36,42,43,...}; so a(3)=a(5)=a(9)=...=a(43)=2, etc.
%e [4]={4}; a(4)=4
%e [6]={6,7,15,16,22,28,59,66,81,...}; a(6)=a(7)=a(15)=...a(81)=6, etc.
%e [8]={8}; a(8)=8
%e [11]={11,18,29,32,35,70,76,...}; a(18)=a(29)=...=a(76)=11, etc.
%e [13]={13,31,39,44,54,60,90,...}; a(31)=a(39)=...=a(90)=13, etc.
%e [14]={14}; a(14)=14
%e [19]={19,33,34,48,...}
%e [21]={21,53,67,69,...}
%e [26]={26,30,37,38,41,47,55,58,65,68,71,95,99,...}
%e [27]={27,62,88,...}
%o (PARI) {A(z) = x=z; y=z; xx=2*x-4; while (y<=xx, x--; xx-=2; if (bittest(y,0)==1, y=x+((y+1)>>1), y=x-(y>>1))); return(x+y-1);};
%o {B(z) = a=z; n=1; while (a!=n, if (a<n, a=2*(n-a), a>2*n, a--, a=2*(a-n)-1);n++); return(a);}; \\ _Lars Blomberg_, Apr 29 2019
%Y Cf. A007063, A006852, A038807, A307797.
%K nonn
%O 1,2
%A _David James Sycamore_, Apr 12 2019
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