|
|
A304337
|
|
Lexicographically earliest fractal-like sequence such that the erasure of all pairs of contiguous terms of opposite parity leaves the sequence unchanged.
|
|
1
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
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.
|
|
LINKS
|
|
|
EXAMPLE
|
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.
|
|
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).
|
|
KEYWORD
|
nonn,base
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|