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

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, 2, 4, 8, 12, 16, 20, 19, 15, 11, 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

Offset: 1

Author

Eric Angelini, May 11 2018

Keywords

• nonn
• base

Comments

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 pairs of parentheses;

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

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;

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.

Examples

			Parentheses are added around each pair of terms of opposite parity:
(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,
Erasing all the parenthesized contents yields
(...),(...),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,
We see that the remaining terms slowly rebuild the starting sequence.
		

Links

• Carole Dubois, Table of n, a(n) for n = 1..5000

Crossrefs

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).