A307340 A fractal 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, 8, 7, 6, 5, 4, 3, 2, 1, 0, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 9, 9, 8, 7, 6, 5, 4, 3, 2, 1, 9, 9, 0, 9, 9, 9, 8, 7, 6, 5, 9, 9, 4, 9, 9, 9, 3, 7, 2, 1, 9, 8, 9, 6, 5, 4, 9, 0, 0, 9, 3, 9, 2, 9, 8, 6, 1, 9, 3, 9, 8, 7, 4, 5, 7, 6, 5, 2, 9, 1, 9, 9, 9, 9, 4, 9, 9, 9, 0, 6, 9, 9, 8, 7, 3, 5, 9, 9, 9, 7, 4, 9

Offset

1,1

Comments

This is defined to be the lexicographically earliest 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); if a(n) = 8, underline a(n+9); if a(n) = 9, underline a(n+10). 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..2007

Example

The sequence starts (9,8,7,6,5,4,3,2,1,0,9,9,9,9,9,9,9,9,9,9,8,7,6,5,4,3,2,1,0,9,9,...)
Instead of underlining terms, we will put parentheses around the terms we want to emphasize:
a(1) = 9 produces parentheses around a(1 + 10 = 11):
9,8,7,6,5,4,3,2,1,0,(9),9,9,9,9,9,9,9,9,9,8,7,6,5,4,3,2,1,0,9,...
a(2) = 8 produces parentheses around a(2 + 9 = 11), which is now already done. Then,
a(3) = 7 produces parentheses around a(3 + 8 = 11), which is already done. Then,
a(4) = 6 produces parentheses around a(4 + 7 = 11) - already done. Then,
a(5) = 5 produces parentheses around a(5 + 6 = 11) - already done. Then,
a(6) = 4 produces parentheses around a(6 + 5 = 11) - already done. Then,
a(7) = 3 produces parentheses around a(7 + 4 = 11) - already done. Then,
a(8) = 2 produces parentheses around a(8 + 3 = 11) - already done. Then,
a(9) = 1 produces parentheses around a(9 + 2 = 11) - already done. Then,
a(10) = 0 produces parentheses around a(10 + 1 = 11) - already done. Now,
a(11) = 9 produces parentheses around a(11 + 10 = 21):
9,8,7,6,5,4,3,2,1,0,(9),9,9,9,9,9,9,9,9,9,(8),7,6,5,4,3,2,1,0,9,...
a(12) = 9 produces parentheses around a(12 + 10 = 22):
9,8,7,6,5,4,3,2,1,0,(9),9,9,9,9,9,9,9,9,9,(8),(7),6,5,4,3,2,1,0,9,...
a(13) = 9 produces parentheses around a(13 + 10 = 23):
9,8,7,6,5,4,3,2,1,0,(9),9,9,9,9,9,9,9,9,9,(8),(7),(6),5,4,3,2,1,0,9,...
a(14) = 9 produces parentheses around a(14 + 10 = 24):
9,8,7,6,5,4,3,2,1,0,(9),9,9,9,9,9,9,9,9,9,(8),(7),(6),(5),4,3,2,1,0,9,...
a(15) = 9 produces parentheses around a(15 + 10 = 25):
9,8,7,6,5,4,3,2,1,0,(9),9,9,9,9,9,9,9,9,9,(8),(7),(6),(5),(4),3,2,1,0,9,...
a(16) = 9 produces parentheses around a(16 + 10 = 26):
9,8,7,6,5,4,3,2,1,0,(9),9,9,9,9,9,9,9,9,9,(8),(7),(6),(5),(4),(3),2,1,0,9,...
a(17) = 9 produces parentheses around a(17 + 10 = 27):
9,8,7,6,5,4,3,2,1,0,(9),9,9,9,9,9,9,9,9,9,(8),(7),(6),(5),(4),(3),(2),1,0,9,...
a(18) = 9 produces parentheses around a(18 + 10 = 28):
9,8,7,6,5,4,3,2,1,0,(9),9,9,9,9,9,9,9,9,9,(8),(7),(6),(5),(4),(3),(2),(1),0,9,...
a(19) = 9 produces parentheses around a(19 + 10 = 29):
9,8,7,6,5,4,3,2,1,0,(9),9,9,9,9,9,9,9,9,9,(8),(7),(6),(5),(4),(3),(2),(1),(0),9,...
a(20) = 9 produces parentheses around a(20 + 10 = 30):
9,8,7,6,5,4,3,2,1,0,(9),9,9,9,9,9,9,9,9,9,(8),(7),(6),(5),(4),(3),(2),(1),(0),(9)...
Etc.
We see in this small example that the parenthesized terms reproduce the initial sequence:
(9),(8),(7),(6),(5),(4),(3),(2),(1),(0),(9)...
The same is true for the subsequence of non-parenthesized terms:
9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 9, 9, 9, 9, 9, 9, 9, 9, 9,...

Crossrefs

Cf. A307183 (first binary example of such fractal sequences), A307332 (ternary), A307333 (quaternary), A307335 (quinary), A307336 (senary), A307337 (septuary), A307338 (octal), A307339 (nonary).

Sequence in context: A022965 A023451 A109910 * A084019 A061601 A098756

Adjacent sequences: A307337 A307338 A307339 * A307341 A307342 A307343

Keyword

base,nonn

Author

Eric Angelini and Carole Dubois, Apr 02 2019

Status

approved