A351330 -id:A351330 - cp's OEIS frontend

A351329 A fractal-like sequence: erase all triples of adjacent terms that have an even sum; the remaining terms rebuild the starting sequence.

1, 2, 3, 4, 6, 8, 1, 2, 10, 5, 7, 3, 9, 11, 12, 4, 13, 14, 15, 6, 8, 1, 2, 10, 5, 16, 18, 20, 7, 22, 24, 26, 3, 28, 30, 32, 9, 34, 36, 38, 11, 12, 4, 13, 14, 40, 17, 19, 15, 21, 23, 42, 6, 25, 44, 27, 8, 46, 29, 31, 1, 33, 35, 48, 2, 37, 50, 39, 10, 52, 41, 43, 5, 45, 47, 54, 16, 49, 56, 51, 18, 20, 7, 22

Offset: 1

Eric Angelini and Carole Dubois, Feb 07 2022

This is the lexicographically earliest such sequence starting with a(1) = 1 and showing no duplicate term in any triple to be erased.
The sequence is fractal-like as it embeds an infinite number of copies of itself.
The sequence was built according to these rules (see, in the Example section, the parenthesization technique):
  1) no overlapping triple of parentheses; a triple is made of integers X, Y and Z;
  2) always start the content inside a pair of parentheses with the smallest integer X > 1 not yet present inside another pair of parentheses and not leading to a contradiction;
  3) always follow X with the smallest integer Y > 1 not yet present inside another pair of parentheses and not leading to a contradiction;
  4) always end the content inside a pair of parentheses with the smallest integer Z > 1 not yet present inside another pair of parentheses and not leading to a contradiction such that X + Y + Z is even;
  5) after a(1) = 1, a(2) = 2 and a(3) = 3, always try to extend the sequence with a duplicate > 2 of the oldest term of the sequence not yet duplicated; if this leads to a contradiction, open a new pair of parentheses.

Eric Angelini, Fabriquons une suite fractale, January 22nd 2022, personal blog (in French).

For other erasing criteria, cf. A303845 (prime by concatenation), A303948 (pair sharing a digit), A274329 (pair summing up to a prime), A351330 (triples having an odd sum).

Parentheses are added around each triple of terms that have an even sum:
(1, 2, 3), (4, 6, 8), 1, 2, (10, 5, 7), 3, (9, 11, 12), 4, (13, 14, 15), 6, 8, 1, 2, 10, 5, (16, 18, 20), 7, (22, 24, 26), 3, (28, 30, 32), 9, (34, 36, 38), 11, 12, 4, 13, 14, (40, 17, 19), 15, (21, 23, 42), 6, (25, 44, 27), 8, (46, 29, 31), 1, ...
Erasing all the parenthesized contents yields
(...), (...), 1, 2, (...), 3, (...), 4, (...), 6, 8, 1, 2, 10, 5, (...), 7, (...), 3, (...), 9, (...), 11, 12, 4, 13, 14, (...), 15, (...), 6, (...), 8, (...), 1, ...
We see that the remaining terms slowly rebuild the starting sequence.

A375244 List of triples {w;x;y} where «w» is the w-th «pyramid», "x" = the number of elements in the «pyramid» that are not erased (before the end-level erasure); "y" is the number of steps in the pyramid until the iteration stops. See the Comments section for more details.

Original entry on oeis.org

1, 13, 12, 2, 12, 11, 3, 12, 10, 4, 11, 10, 5, 2, 4, 6, 11, 9, 7, 127, 79, 8, 10, 9, 9, 9, 7, 10, 1, 3, 11, 1, 2, 12, 10, 8, 13, 10, 8, 14, 126, 78, 15, 121, 77, 16, 9, 8, 17, 119, 75, 18, 8, 6, 19, 118, 74, 20, 1, 3, 21, 136, 76, 22, 1, 2, 23, 134, 75, 24, 120, 74, 25, 9, 9, 26, 121, 74, 27, 7, 8, 28, 116, 73

Offset: 1

Eric Angelini and Jean-Marc Falcoz, Aug 07 2024

Start the top of a "pyramid" with an integer w.
Form the lower level by adding w to each digit of w.
Erase any term having one or more duplicates, as well as its duplicates.
Iterate.
All "pyramids" will be blocked at some point, because their lowest level will end up completely erased.

			We start the first pyramid with w = 1:
.
                                 1
                                 2
                                 4
                                 8
                                16
                               17.22
                            18.24.24.24
             (a triple erasure, 22 will be erased later)
                               19.26
                            20.28.28.32
             (a double erasure, 20 will be erased later)
                            22.20.35.34
              (we erase now the 22-pair and the 20-pair)
                            38.40.37.38
      (a double erasure again, 40 will be erased at the next step)
                            44.40.40.44
.
The iteration stops there. w = 1, x = 13 as 13 terms were not erased in the blocked pyramid, y = 12 as the now blocked-pyramid has 12 levels.
Those numbers form the first triple of the sequence {1;13;12}.

Eric Angelini, This is not a fractal pipe, Personal blog of the author.

Cf. A351330.

cp's OEIS Frontend

A351329 A fractal-like sequence: erase all triples of adjacent terms that have an even sum; the remaining terms rebuild the starting sequence.

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