Illustration of the relationship between A088643, A132075 and A255312.

                                        Peter Munn, August 2021

1. How A132075 is constructed from A088643
==========================================

The first table below illustrates the construction.

The 2nd column shows the rows of of A088643 reversed and truncated to contain floor(n/4) terms -- these are the terms used to construct A132075.

The column entitled "match?" shows
(1) "=" if the truncated reversed row n of A088643 (the 2nd column) matches the terms of A132075 that have been established by earlier rows of A088643; otherwise
(2) a number, m, followed by "<>" to indicate the value of the first term in the 2nd column that does not match the established terms of A132075.

If the terms in the 2nd column match the established terms of A132075 ("=" appears in the match column): 
(1) if there are more terms in the 2nd column than established terms, then the match column is followed by "->" with the 2nd column repeated in the final column -- so these terms extend the established terms; otherwise
(2) "(not longer)" appears in the final column.

   n   truncated reversed rows of    match?   established terms of
       A088643 (used for A132075)             A132075 after row n
 -----------------------------------------------------------------
                                             1 2 3 (by definition)
   1   (null)                         =       (not longer)
   2   (null)                         =       (not longer)
   3   (null)                         =       (not longer)
   4   1                              =       (not longer)
   5   1                              =       (not longer)
   6   1                              =       (not longer)
   7   1                              =       (not longer)
   8   1 2                            =       (not longer)
   9   1 2                            =       (not longer)
  10   1 2                            =       (not longer)
  11   1 4                            4<>            
  12   1 4 3                          4<>            
  13   1 4 3                          4<>            
  14   1 2 5                          5<>            
  15   1 2 5                          5<>            
  16   1 2 5 8                        5<>            
  17   1 4 3 2                        4<>            
  18   1 4 3 2                        4<>            
  19   1 4 3 2                        4<>            
  20   1 4 3 2 5                      4<>            
  21   1 4 3 2 5                      4<>            
  22   1 4 3 2 5                      4<>            
  23   1 4 3 2 5                      4<>            
  24   1 4 3 2 5 6                    4<>            
  25   1 4 3 2 5 6                    4<>            
  26   1 2 3 4 7 6                    =   ->  1 2 3 4 7 6     
  27   1 2 3 4 7 6                    =        (not longer)
  28   1 4 3 2 5 6 7                  4<>            
  29   1 4 3 2 5 6 7                  4<>            
  30   1 4 3 2 5 6 7                  4<>            
  31   1 4 3 2 5 6 7                  4<>            
  32   1 4 3 2 5 6 7 10               4<>            
  33   1 2 3 4 7 6 5 14               =   ->  1 2 3 4 7 6 5 14   
  34   1 2 3 4 7 6 5 14               =        (not longer)
  35   1 2 3 4 7 6 5 14               =        (not longer)
  36   1 2 3 4 7 6 5 14  9            =   ->  1 2 3 4 7 6 5 14 9  
  37   1 2 3 4 7 6 5 14  9            =        (not longer)
  38   1 2 3 4 7 6 5 14  9            =        (not longer)
  39   1 2 3 4 7 6 5 14  9            =        (not longer)
  40   1 2 3 4 7 6 5 14  9 10         =   ->  1 2 3 4 7 6 5 14 9 10 
  41   1 2 3 4 7 6 5 14  9 10         =        (not longer)
  42   1 2 3 4 7 6 5 14  9 10         =        (not longer)
  43   1 4 3 2 5 6 7 10  9  8         4<>            
  44   1 2 3 4 7 6 5 14  9 10 13      =   ->  1 2 3 4 7 6 5 14 9 10 13
  45   1 2 3 4 7 6 5 14  9 10 13      =        (not longer)
  46   1 2 3 4 7 6 5 14  9 10 13      =        (not longer)
  47   1 2 3 4 7 6 5 14  9 10 13      =        (not longer)
  48   1 4 3 2 5 6 7 10  9  8 11 12   4<>            
  49   1 4 3 2 5 6 7 10  9  8 11 12   4<>            
  50   1 4 3 2 5 6 7 10  9  8 11 12   4<>            
  ...
A132075 for comparison:
       1 2 3 4 7 6 5 14  9 10 13 16 ...


2. The "hindsight" view: A255312
================================

Having seen how A132075 is built, we can examine the relationship with A255312. A255312 compares the full rows of A088643 with A132075 _after_ we have found sufficient terms of A132075 to do a complete comparison. In a sense, A255312 gives a "hindsight" view.

The modified table below shows this relationship with A255312.

The column entitled "match?" now shows
(1) "==" if the full reversed row n of A088643 matches the initial n terms of A132075 (the condition for n to be listed in A255312); otherwise
(2) "=|" if the truncated reversed row n of A088643 (the 2nd column) matches the terms of A132075 that have been established by earlier rows of A088643; otherwise (as in the first table)
(3) a number, m, followed by "<>" to indicate the value of the first term in the 2nd column that does not match the established terms of A132075.

Note particularly the rows that show "==      (not longer)": this appears when the full reversed row A088643 is seen later (in "hindsight") to match the start of the full A132075 sequence, but the row does not contribute to extending A132075 because it was not long enough when truncated.
On the other hand when we see "=|  ->", this indicates a reversed row that contributes to extending A132075 -- because, when truncated, it matches the established start of A132075 -- but during the subsequent process of extending A132075, A132075 ceases to match the remainder of the reversed row. For example, A132075 matches only the first 27 terms of the reversed row 33 of A088643.

   n   truncated reversed rows of    match?   established terms of
       A088643 (used for A132075)             A132075 after row n
 -----------------------------------------------------------------
                                             1 2 3 (by definition)
   1   (null)                         ==      (not longer)
   2   (null)                         ==      (not longer)
   3   (null)                         ==      (not longer)
   4   1                              ==      (not longer)
   5   1                              =|      (not longer)
   6   1                              =|      (not longer)
   7   1                              =|      (not longer)
   8   1 2                            =|      (not longer)
   9   1 2                            =|      (not longer)
  10   1 2                            =|      (not longer)
  11   1 4                            4<>            
  12   1 4 3                          4<>            
  13   1 4 3                          4<>            
  14   1 2 5                          5<>            
  15   1 2 5                          5<>            
  16   1 2 5 8                        5<>            
  17   1 4 3 2                        4<>            
  18   1 4 3 2                        4<>            
  19   1 4 3 2                        4<>            
  20   1 4 3 2 5                      4<>            
  21   1 4 3 2 5                      4<>            
  22   1 4 3 2 5                      4<>            
  23   1 4 3 2 5                      4<>            
  24   1 4 3 2 5 6                    4<>            
  25   1 4 3 2 5 6                    4<>            
  26   1 2 3 4 7 6                    ==  ->  1 2 3 4 7 6     
  27   1 2 3 4 7 6                    ==       (not longer)
  28   1 4 3 2 5 6 7                  4<>            
  29   1 4 3 2 5 6 7                  4<>            
  30   1 4 3 2 5 6 7                  4<>            
  31   1 4 3 2 5 6 7                  4<>            
  32   1 4 3 2 5 6 7 10               4<>            
  33   1 2 3 4 7 6 5 14               =|  ->  1 2 3 4 7 6 5 14   
  34   1 2 3 4 7 6 5 14               =|       (not longer)
  35   1 2 3 4 7 6 5 14               ==       (not longer)
  36   1 2 3 4 7 6 5 14  9            ==  ->  1 2 3 4 7 6 5 14 9  
  37   1 2 3 4 7 6 5 14  9            ==       (not longer)
  38   1 2 3 4 7 6 5 14  9            =|       (not longer)
  39   1 2 3 4 7 6 5 14  9            =|       (not longer)
  40   1 2 3 4 7 6 5 14  9 10         =|  ->  1 2 3 4 7 6 5 14 9 10 
  41   1 2 3 4 7 6 5 14  9 10         =|       (not longer)
  42   1 2 3 4 7 6 5 14  9 10         =|       (not longer)
  43   1 4 3 2 5 6 7 10  9  8         4<>            
  44   1 2 3 4 7 6 5 14  9 10 13      =|  ->  1 2 3 4 7 6 5 14 9 10 13
  45   1 2 3 4 7 6 5 14  9 10 13      =|       (not longer)
  46   1 2 3 4 7 6 5 14  9 10 13      =|       (not longer)
  47   1 2 3 4 7 6 5 14  9 10 13      ==       (not longer)
  48   1 4 3 2 5 6 7 10  9  8 11 12   4<>            
  49   1 4 3 2 5 6 7 10  9  8 11 12   4<>            
  50   1 4 3 2 5 6 7 10  9  8 11 12   4<>            
  ...
A132075 for comparison:
       1 2 3 4 7 6 5 14  9 10 13 16 ...