A304337 - OEIS

%I #13 Feb 05 2022 16:24:18

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

%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 pairs of parentheses;

%C   2) always start the content inside a pair of parentheses with the smallest integer X not yet present inside another pair of parentheses;

%C   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   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 Carole Dubois, <a href="/A304337/b304337.txt">Table of n, a(n) for n = 1..5000</a>

%e Parentheses are added around each pair of terms of opposite parity:

%e (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 Erasing all the parenthesized contents yields

%e (...),(...),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 We see that the remaining terms slowly rebuild the starting sequence.

%Y 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 nonn,base

%O 1,2

%A _Eric Angelini_, May 11 2018