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A307335 A fractal quinary sequence: For all n >= 1, underline the term with index n + a(n) + 1; then the two subsequences of underlined terms and of non-underlined terms are both equal to the sequence itself. 9
4, 3, 2, 1, 0, 4, 4, 4, 4, 4, 3, 2, 1, 0, 4, 4, 3, 2, 1, 4, 4, 0, 3, 4, 4, 4, 2, 4, 1, 0, 4, 3, 4, 2, 1, 3, 1, 2, 4, 4, 0, 4, 4, 3, 4, 4, 4, 2, 0, 4, 1, 0, 3, 3, 4, 4, 4, 2, 4, 1, 3, 1, 4, 2, 4, 2, 0, 4, 4, 4, 4, 1, 3, 4, 4, 4, 2, 0, 0, 4, 1, 4, 3, 3, 0, 4, 3, 4, 4, 2, 4, 1, 2, 4, 3, 1, 1, 1, 3, 4, 2, 4, 2, 2, 4, 4, 0, 4, 4, 4, 4, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
This is defined to be the lexicographically earliest quinary sequence with the following property:
If a(n) = 0, underline a(n+1); if a(n) = 1, underline a(n+2); if a(n) = 2, underline a(n+3); if a(n) = 3, underline a(n+4); if a(n) = 4, underline a(n+5). Now, the subsequence of (once or more) underlined terms must be equal to the original sequence (copy #1), and the subsequence of non-underlined terms must also reproduce the original sequence (copy #2).
LINKS
EXAMPLE
The sequence starts (4,3,2,1,0,4,4,4,4,4,3,2,1,0,4,4,3,2,1,4,4,...)
Instead of underlining terms, we will put parentheses around the terms we want to emphasize:
a(1) = 4 produces parentheses around a(1 + 5 = 6):
4,3,2,1,0,(4),4,4,4,4,3,2,1,0,4,4,3,2,1,4,4,...
a(2) = 3 produces parentheses around a(2 + 4 = 6), which is now already done. Then,
a(3) = 2 produces parentheses around a(3 + 3 = 6), which is already done. Then,
a(4) = 1 produces parentheses around a(4 + 2 = 6), - already done. Then,
a(5) = 0 produces parentheses around a(5 + 1 = 6) - already done. Then,
a(6) = 4 produces parentheses around a(6 + 5 = 11):
4,3,2,1,0,(4),4,4,4,4,(3),2,1,0,4,4,3,2,1,4,4,...
a(7) = 4 produces parentheses around a(7 + 5 = 12):
4,3,2,1,0,(4),4,4,4,4,(3),(2),1,0,4,4,3,2,1,4,4,...
a(8) = 4 produces parentheses around a(8 + 5 = 13):
4,3,2,1,0,(4),4,4,4,4,(3),(2),(1),0,4,4,3,2,1,4,4,...
a(9) = 4 produces parentheses around a(9 + 5 = 14):
4,3,2,1,0,(4),4,4,4,4,(3),(2),(1),(0),4,4,3,2,1,4,4,...
a(10) = 4 produces parentheses around a(10 + 5 = 15):
4,3,2,1,0,(4),4,4,4,4,(3),(2),(1),(0),(4),4,3,2,1,4,4,...
a(11) = 3 produces parentheses around a(11 + 4 = 15) - already done. Then,
a(12) = 2 produces parentheses around a(12 + 3 = 15) - already done. Then,
a(13) = 1 produces parentheses around a(13 + 2 = 15) - already done. Then,
a(14) = 0 produces parentheses around a(14 + 1 = 15) - already done. Then,
a(15) = 4 produces parentheses around a(15 + 5 = 20):
4,3,2,1,0,(4),4,4,4,4,(3),(2),(1),(0),(4),4,3,2,1,(4),4,...
a(16) = 4 produces parentheses around a(16 + 5 = 21):
4,3,2,1,0,(4),4,4,4,4,(3),(2),(1),(0),(4),4,3,2,1,(4),(4),... Etc.
We see in this small example that the parenthesized terms reproduce the initial sequence:
(4),(3),(2),(1),(0),(4),(4),(4)...
The same is true for the subsequence of non-parenthesized terms:
4, 3, 2, 1, 0, 4, 4, 4, 4, 4, 3, 2, 1,...
CROSSREFS
Cf. A307183 (first binary example of such fractal sequences), A307332 (ternary), A307333 (quaternary), A307336 (senary), A307337 (septuary), A307338 (octal), A307339 (nonary), A307340 (decimal).
Sequence in context: A066440 A182425 A071692 * A030586 A249345 A070635
KEYWORD
base,nonn
AUTHOR
Eric Angelini and Carole Dubois, Apr 02 2019
STATUS
approved

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Last modified February 22 09:35 EST 2024. Contains 370250 sequences. (Running on oeis4.)