A307338 A fractal octal (8 elements) 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.

7, 6, 5, 4, 3, 2, 1, 0, 7, 7, 7, 7, 7, 7, 7, 7, 6, 5, 4, 3, 2, 1, 0, 7, 7, 6, 5, 4, 3, 2, 1, 7, 7, 0, 7, 7, 7, 6, 5, 7, 7, 4, 7, 7, 6, 3, 3, 5, 4, 0, 2, 1, 2, 7, 1, 7, 5, 7, 7, 0, 7, 6, 3, 4, 7, 2, 1, 7, 7, 7, 6, 5, 0, 6, 7, 7, 7, 7, 7, 4, 5, 7, 7, 7, 4, 7, 7, 7, 6, 6, 3, 3, 3, 5, 4, 0, 2, 3, 5, 1, 4, 2, 0, 7, 7, 1, 2, 7, 1, 0, 6

Offset

1,1

Comments

This is defined to be the lexicographically earliest octal 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); if a(n) = 5, underline a(n+6); if a(n) = 6, underline a(n+7); if a(n) = 7, underline a(n+8). 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 (7,6,5,4,3,2,1,0,7,7,7,7,7,7,7,7,6,5,4,3,2,1,0,7,7,...)
Instead of underlining terms, we will put parentheses around the terms we want to emphasize:
a(1) = 7 produces parentheses around a(1 + 8 = 9):
7,6,5,4,3,2,1,0,(7),7,7,7,7,7,7,7,6,5,4,3,2,1,0,7,...
a(2) = 6 produces parentheses around a(2 + 7 = 9), which is now already done. Then,
a(3) = 5 produces parentheses around a(3 + 6 = 9), which is already done. Then,
a(4) = 4 produces parentheses around a(4 + 5 = 9) - already done. Then,
a(5) = 3 produces parentheses around a(5 + 4 = 9) - already done. Then,
a(6) = 2 produces parentheses around a(6 + 3 = 9) - already done. Then,
a(7) = 1 produces parentheses around a(7 + 2 = 9) - already done. Then,
a(8) = 0 produces parentheses around a(8 + 1 = 9) - already done. Then,
a(9) = 7 produces parentheses around a(9 + 8 = 17):
7,6,5,4,3,2,1,0,(7),7,7,7,7,7,7,7,(6),5,4,3,2,1,0,7,...
a(10) = 7 produces parentheses around a(10 + 8 = 18):
7,6,5,4,3,2,1,0,(7),7,7,7,7,7,7,7,(6),(5),4,3,2,1,0,7,...
a(11) = 7 produces parentheses around a(11 + 8 = 19):
7,6,5,4,3,2,1,0,(7),7,7,7,7,7,7,7,(6),(5),(4),3,2,1,0,7,...
a(12) = 7 produces parentheses around a(12 + 8 = 20):
7,6,5,4,3,2,1,0,(7),7,7,7,7,7,7,7,(6),(5),(4),(3),2,1,0,7,...
a(13) = 7 produces parentheses around a(13 + 8 = 21):
7,6,5,4,3,2,1,0,(7),7,7,7,7,7,7,7,(6),(5),(4),(3),(2),1,0,7,...
a(14) = 7 produces parentheses around a(14 + 8 = 22):
7,6,5,4,3,2,1,0,(7),7,7,7,7,7,7,7,(6),(5),(4),(3),(2),(1),0,7,...
a(15) = 7 produces parentheses around a(15 + 8 = 23):
7,6,5,4,3,2,1,0,(7),7,7,7,7,7,7,7,(6),(5),(4),(3),(2),(1),(0),7,...
a(16) = 7 produces parentheses around a(16 + 8 = 24):
7,6,5,4,3,2,1,0,(7),7,7,7,7,7,7,7,(6),(5),(4),(3),(2),(1),(0),(7)...
a(17) = 6 produces parentheses around a(17 + 7 = 24) - already done. Then,
a(18) = 5 produces parentheses around a(18 + 6 = 24) - already done. Then,
a(19) = 4 produces parentheses around a(19 + 5 = 24) - already done. Etc.
We see in this small example that the parenthesized terms reproduce the initial sequence:
(7),(6),(5),(4),(3),(2),(1),(0),(7)...
The same is true for the subsequence of non-parenthesized terms:
7, 6, 5, 4, 3, 2, 1, 0, 7, 7, 7, 7, 7, 7, 7,...

Crossrefs

Cf. A307183 (first binary example of such fractal sequences), A307332 (ternary), A307333 (quaternary), A307335 (quinary), A307336 (senary), A307337 (septuary), A307339 (nonary), A307340 (decimal).

Sequence in context: A072558 A022963 A023449 * A031098 A004448 A284807

Adjacent sequences: A307335 A307336 A307337 * A307339 A307340 A307341

Keyword

base,nonn

Author

Eric Angelini and Carole Dubois, Apr 02 2019

Status

approved