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%I #6 Feb 07 2022 21:44:16
%S 1,2,4,6,8,3,1,2,5,7,9,4,11,13,15,6,17,19,21,8,3,1,2,5,7,10,23,12,9,
%T 25,14,16,4,18,20,27,11,22,29,24,13,15,6,17,19,26,31,28,21,33,30,32,8,
%U 34,36,35,3,38,37,40,1,39,42,44,2,46,48,41,5,50,43,52,7,45,54,56,10,58,60,47,23,12,9,25,14
%N A fractal-like sequence: erase all triples of contiguous terms that have an odd sum; the remaining terms rebuild the starting sequence.
%C This is the lexicographically earliest such sequence starting with a(1) = 1 and showing no duplicate term in any triple to be erased.
%C The sequence is fractal-like as it embeds an infinite number of copies of itself.
%C The sequence was built according to these rules (see, in the Example section, the parenthesization technique):
%C 1) no overlapping triple of parentheses; a triple is made of integers X, Y and Z;
%C 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;
%C 3) always follow X with the smallest integer Y > 1 not yet present inside another pair of parentheses and not leading to a contradiction;
%C 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 odd;
%C 5) after a(1) = 1, a(2) = 2 and a(3) = 4, 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.
%H Eric Angelini, <a href="http://cinquantesignes.blogspot.com/2022/01/fabriquons-une-suite-fractale.html">Fabriquons une suite fractale</a>, January 22nd 2022, personal blog (in French).
%e Parentheses are added around each triple of terms that have an odd sum:
%e (1, 2, 4), (6, 8, 3), 1, 2, (5, 7, 9), 4, (11, 13, 15), 6, (17, 19, 21), 8, 3, 1, 2, 5, 7, (10, 23, 12), 9, (25, 14, 16), 4, (18, 20, 27), 11, (22, 29, 24), 13, 15, 6, 17, 19, (26, 31, 28), 21, (33, 30, 32), 8, (34, 36, 35), 3, (38, 37, 40), 1, (39, 42, 44), 2,...
%e Erasing all the parenthesized contents yields
%e (...), (...), 1, 2, (...), 4, (...), 6, (...), 8, 3, 1, 2, 5, 7, (...), 9, (...), 4, (...), 11, (...), 13, 15, 6, 17, 19, (...), 21, (...), 8, (...), 3, (...), 1, (...), 2,...
%e We see that the remaining terms slowly rebuild the starting sequence.
%Y For other erasing criteria, cf. A303845 (prime by concatenation), A303948 (pair sharing a digit), A274329 (pair summing up to a prime), A351329 (triples having an even sum).
%K base,nonn
%O 1,2
%A _Eric Angelini_ and _Carole Dubois_, Feb 07 2022
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