A307333 A fractal quaternary 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.

3, 2, 1, 0, 3, 3, 3, 3, 2, 1, 0, 3, 3, 2, 1, 3, 3, 0, 1, 3, 2, 3, 3, 0, 2, 3, 3, 1, 2, 3, 3, 0, 2, 1, 3, 3, 1, 3, 3, 0, 3, 2, 3, 3, 1, 0, 2, 3, 1, 3, 2, 0, 3, 1, 3, 1, 3, 3, 3, 2, 0, 3, 2, 3, 3, 3, 0, 3, 1, 0, 3, 2, 2, 3, 1, 3, 0, 2, 3, 3, 1, 1, 3, 1, 2, 3, 3, 3, 3, 2, 0, 3, 2, 3, 0, 3, 2, 3, 1, 3, 3, 0, 3, 1, 0, 2, 2, 3, 3, 1, 3, 3, 0

Offset

1,1

Comments

This is defined to be the lexicographically earliest quaternary 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). 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..2004

Example

The sequence starts (3, 2, 1, 0, 3, 3, 3, 3, 2, 1, 0, 3, 3, 2, 1, 3, 3, 0,...)
Instead of underlining terms, we will put parentheses around the terms we want to emphasize:
a(1) = 3 produces parentheses around a(1 + 4 = 5):
3, 2, 1, 0, (3,) 3, 3, 3, 2, 1, 0, 3, 3, 2, 1, 3, 3, 0,...
a(2) = 2 produces parentheses around a(2 + 3 = 5), which is now already done. Then,
a(3) = 1 produces parentheses around a(3 + 2 = 5), which is already done. Then,
a(4) = 0 produces parentheses around a(4 + 1 = 5), which is already done. Now,
a(5) = 3 produces parentheses around a(5 + 4 = 9):
3, 2, 1, 0, (3,) 3, 3, 3, (2), 1, 0, 3, 3, 2, 1, 3, 3, 0,...
a(6) = 3 produces parentheses around a(6 + 4 = 10):
3, 2, 1, 0, (3,) 3, 3, 3, (2), (1), 0, 3, 3, 2, 1, 3, 3, 0,...
a(7) = 3 produces parentheses around a(7 + 4 = 11):
3, 2, 1, 0, (3,) 3, 3, 3, (2), (1), (0), 3, 3, 2, 1, 3, 3, 0,...
a(8) = 3 produces parentheses around a(8 + 4 = 12):
3, 2, 1, 0, (3,) 3, 3, 3, (2), (1), (0), (3), 3, 2, 1, 3, 3, 0,...
a(9) = 2 produces parentheses around a(9 + 3 = 12) - already done. Then,
a(10) = 1 produces parentheses around a(10 + 2 = 12) - already done. Then,
a(11) = 0 produces parentheses around a(11 + 1 = 12) - already done. Then,
a(12) = 3 produces parentheses around a(12 + 4 = 16):
3, 2, 1, 0, (3,) 3, 3, 3, (2), (1), (0), (3), 3, 2, 1, (3), 3, 0,...
a(13) = 3 produces parentheses around a(13 + 4 = 17):
3, 2, 1, 0, (3,) 3, 3, 3, (2), (1), (0), (3), 3, 2, 1, (3), (3), 0,...
a(14) = 2 produces parentheses around a(14 + 3 = 17) - already done. Then,
a(15) = 1 produces parentheses around a(15 + 2 = 17) - already done. Etc.
We see in this small example that the parenthesized terms reproduce the initial sequence:
(3),(2),(1),(0),(3),(3),(3),...
The same is true for the subsequence of non-parenthesized terms:
3, 2, 1, 0, 3, 3, 3, 3, 2, 1, 0,...

Crossrefs

Cf. A307183 (first binary example of such fractal sequences), A307332 (ternary), A307335 (quinary), A307336 (senary), A307337 (septuary), A307338 (octal), A307339 (nonary), A307340 (decimal).

Sequence in context: A083721 A158459 A319666 * A031251 A194885 A128317

Adjacent sequences: A307330 A307331 A307332 * A307334 A307335 A307336

Keyword

base,nonn

Author

Eric Angelini and Carole Dubois, Apr 02 2019

Status

approved