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

1,2

Comments

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.

Links

Table of n, a(n) for n=1..84.

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

Formula

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.

Crossrefs

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).

Sequence in context: A171249 A036413 A069912 * A152306 A120817 A327887

Adjacent sequences: A351326 A351327 A351328 * A351330 A351331 A351332

Keyword

base,nonn

Author

Eric Angelini and Carole Dubois, Feb 07 2022

Status

approved