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A307407
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Irregular table read by rows: rows list terms that map to the nodes in the graph of the "3x+1" (or Collatz) problem.
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3
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16, 4, 5, 1, 10, 2, 3, 40, 12, 13, 64, 20, 21, 88, 28, 29, 9, 58, 112, 36, 37, 136, 44, 45, 160, 52, 53, 17, 106, 34, 35, 11, 70, 22, 23, 7, 46, 14, 15, 184, 60, 61, 208, 68, 69, 232, 76, 77, 25, 154, 50, 51, 256, 84, 85, 280, 92, 93
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OFFSET
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1,1
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COMMENTS
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The construction is similar to that in A322469. The sequence is the flattened form of an irregular table S(i, j) (see the example below) which has rows i >= 1 consisting of subsequences of varying length.
Like Truemper (cf. link), we denote the mapping x -> 2*x by "m" ("multiply"), the mapping x -> (x - 1)/3 by "d" ("divide"), and the combined mapping "dm" x -> (x - 1)/3 * 2 by "s" ("squeeze"). The d mapping is defined only if it is possible, that is, if x - 1 is divisible by 3. We write m, d and s as infix operation words, for example "4 mms 10", and we use exponents for repeated operations, for example "mms^2 = mmss".
Row i in table S is constructed by the following algorithm: Start with 6 * i - 2 in column j = 1. The following columns j are defined in groups of four by the operations:
k j=4*k+2 j=4*k+3 j=4*k+4 j=4*k+5
--------------------------------------------------
0 mm dmm mmd dmmd
1 mms dmms mmsd dmmsd
2 mms^2 dmms^2 mms^2d dmms^2d
...
k mms^k dmms^k mm(s^k)d dmm(s^k)d
The construction for the row terminates at the first column where a d operation is no longer possible. This point is always reached. This can be proved by the observation that, for any row i in S, there is a unique mapping x -> (x + 2)/6 of the terms in column 1, 2, 5, 9, 13, ... 4*m+1 to the terms in row i of table T in A322469. The row construction process in A322469 stops, therefore it stops also in the sequence defined here.
Conjecture: The sequence is a permutation of the positive numbers.
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LINKS
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EXAMPLE
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Table S(i, j) begins:
i\j 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
----------------------------------------------------------------
1: 16 4 5 1 10 2 3
2: 40 12 13
3: 64 20 21
4: 88 28 29 9 58
5: 112 36 37
6: 136 44 45
7: 160 52 53 17 106 34 35 11 70 22 23 7 46 14 15
8: 184 60 61
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PROG
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(Perl) cf. link.
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CROSSREFS
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KEYWORD
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nonn,easy,tabf
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AUTHOR
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STATUS
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approved
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