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A307536
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Self referencing version of the "Kimberling shuffle" sequence (see Comments).
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1
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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, 26, 13, 11, 19, 19, 11, 2, 26, 26, 13, 40, 26, 2, 2, 13, 45, 2, 26, 19, 49, 50, 51, 51, 21, 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
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OFFSET
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1,2
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COMMENTS
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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.
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.
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,...
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.)
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.
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.
The DATA is based on iterating A and B until the value > 10^8. (End)
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REFERENCES
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R. K. Guy, Unsolved Problems Number Theory, Sect E35.
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LINKS
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C. Kimberling, Problem 1615, Crux Mathematicorum, Vol. 17 (2) 44 1991 and Vol. 18, March 1992, pp. 82-83.
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EXAMPLE
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Examples of [k] for the above list up to k=27:
[1]={1}; so a(1)=1
[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.
[4]={4}; a(4)=4
[6]={6,7,15,16,22,28,59,66,81,...}; a(6)=a(7)=a(15)=...a(81)=6, etc.
[8]={8}; a(8)=8
[11]={11,18,29,32,35,70,76,...}; a(18)=a(29)=...=a(76)=11, etc.
[13]={13,31,39,44,54,60,90,...}; a(31)=a(39)=...=a(90)=13, etc.
[14]={14}; a(14)=14
[19]={19,33,34,48,...}
[21]={21,53,67,69,...}
[26]={26,30,37,38,41,47,55,58,65,68,71,95,99,...}
[27]={27,62,88,...}
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PROG
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(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); };
{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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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