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.

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

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

Carole Dubois, Table of n, a(n) for n = 1..2005

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

Adjacent sequences: A307332 A307333 A307334 * A307336 A307337 A307338

Keyword

base,nonn

Author

Eric Angelini and Carole Dubois, Apr 02 2019

Status

approved