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A304337 Lexicographically earliest fractal-like sequence such that the erasure of all pairs of contiguous terms of opposite parity leaves the sequence unchanged.

%I A304337 #13 Feb 05 2022 16:24:18
%S A304337 1,2,4,3,1,5,6,2,4,8,7,3,1,5,9,10,6,2,4,8,12,11,7,3,1,5,9,13,14,10,6,
%T A304337 2,4,8,12,16,15,11,7,3,1,5,9,13,17,18,14,10,6,2,4,8,12,16,20,19,15,11,
%U A304337 7,3,1,5,9,13,17,21,22,18,14,10,6,2,4,8,12,16,20,24,23,19,15,11,7,3,1
%N A304337 Lexicographically earliest fractal-like sequence such that the erasure of all pairs of contiguous terms of opposite parity leaves the sequence unchanged.
%C A304337 The sequence is fractal-like as it embeds an infinite number of copies of itself.
%C A304337 The sequence was built according to these rules (see, in the Example section, the parenthesization technique):
%C A304337   1) no overlapping pairs of parentheses;
%C A304337   2) always start the content inside a pair of parentheses with the smallest integer X not yet present inside another pair of parentheses;
%C A304337   3) always end the content inside a pair of parentheses with the smallest integer Y not yet present inside another pair of parentheses such Y is not of the same parity as X;
%C A304337   4) after a(1) = 1 and a(2) = 2, always try to extend the sequence with a duplicate of the oldest term of the sequence not yet duplicated; if this leads to a contradiction, open a new pair of parentheses.
%H A304337 Carole Dubois, <a href="/A304337/b304337.txt">Table of n, a(n) for n = 1..5000</a>
%e A304337 Parentheses are added around each pair of terms of opposite parity:
%e A304337 (1,2),(4,3),1,(5,6),2,4,(8,7),3,1,5,(9,10),6,2,4,8,(12,11),7,3,1,5,9,(13,14),10,6,2,4,8,12,(16,15),11,7,3,1,5,9,13,(17,18),14,10,6,
%e A304337 Erasing all the parenthesized contents yields
%e A304337 (...),(...),1,(...),2,4,(...),3,1,5,(....),6,2,4,8,(.....),7,3,1,5,9,(.....),10,6,2,4,8,12,(.....),11,7,3,1,5,9,13,(.....),14,10,6,
%e A304337 We see that the remaining terms slowly rebuild the starting sequence.
%Y A304337 Cf. A303845 (same idea, but pairs of contiguous terms are erased if a prime by concatenation arises), A303948 (if pair has at least one digit in common), A303953 (if pair sums up to a square).
%K A304337 nonn,base
%O A304337 1,2
%A A304337 _Eric Angelini_, May 11 2018