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Created page with "===Name=== Irregular table of rows with terms which map uniquely to the nodes in the graph of the "3x+1" (or Collatz) problem. ===Data=== 16, 4, 5, 1, 10, 2, 3, 40, 12, 13, 6..."

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===Name===
Irregular table of rows with terms which map uniquely to the nodes in the graph of the "3x+1" (or Collatz) problem.
===Data===
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
===Offset 1,1===
===Comments===
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. We define three mappings:
d (for "divide") maps n -> (n - 1)/3 if n == 1 mod 3,
m (for "multiply") maps n = n/2 for all n,
s (for "squeeze") is dm and maps n -> ((n - 1) / 3) * 2.
We note the mappings as infix operators, for example 16 d 5 and 5 m 10. For any positive row number i (also the segment "index" or "left side") as starting value, we compute the elements S[i, j] by the mappings shown in the column "Map" of the following table:
j | Map | Form of i | S(i, j) | Residues | Residues not yet covered
--+------+ -------------+--------------+------------+-------------------------
1 | | 1 * k + 1 | 4 * k + 3 | 3 mod 4 | 0, 1, 2 mod 4
2 | / 3 | 3 * k + 1 | 4 * k + 1 | 1 mod 4 | 0, 2, 4, 6 mod 8
3 | * 2 | 3 * k + 1 | 8 * k + 2 | 2 mod 8 | 0, 4, 6 mod 8
4 | / 3 | 9 * k + 7 | 8 * k + 6 | 6 mod 8 | 0, 4, 8, 12 mod 16
5 | * 2 | 9 * k + 7 | 16 * k + 12 | 12 mod 16 | 0, 4, 8 mod 16
6 | / 3 | 27 * k + 7 | 16 * k + 4 | 4 mod 16 | 0, 8, 16, 24 mod 32
7 | * 2 | 27 * k + 7 | 32 * k + 8 | 8 mod 32 | 0, 16, 24 mod 32
8 | / 3 | 81 * k + 61 | 32 * k + 24 | 24 mod 32 | 0, 16, 32, 48 mod 64
9 | * 2 | 81 * k + 61 | 64 * k + 48 | 48 mod 64 | 0, 16, 32 mod 64
..| ... | e * k + f | g * k + m | m mod g | 0, ...

the following series of mappings:
j = 1: mm, j 2 as defined by the following algorithm:
Construction and proof are similiar to that in A322469, which is more simple.
===Example===
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
===Crossrefs===

OEIS/A307407 - Revision history

imported>Gfis: Created page with "===Name=== Irregular table of rows with terms which map uniquely to the nodes in the graph of the "3x+1" (or Collatz) problem. ===Data=== 16, 4, 5, 1, 10, 2, 3, 40, 12, 13, 6..."