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A307339 A fractal nonary 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, 8, 8, 8, 8, 8, 8, 8, 8, 8, 7, 6, 5, 4, 3, 2, 1, 0, 8, 8, 7, 6, 5, 4, 3, 2, 1, 8, 8, 0, 8, 8, 8, 7, 6, 5, 8, 8, 4, 8, 8, 8, 3, 5, 2, 7, 6, 1, 4, 3, 2, 1, 8, 0, 8, 8, 0, 5, 8, 8, 8, 8, 7, 7, 6, 6, 5, 4, 3, 2, 1, 8, 8, 8, 8, 4, 8, 8, 8, 3, 0, 8, 8, 8, 5, 7, 6, 5, 2, 7, 8, 8, 4, 8, 6, 1, 4, 8, 3, 8, 3, 2, 5 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

This is defined to be the lexicographically earliest nonary 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). 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 (8,7,6,5,4,3,2,1,0,8,8,8,8,8,8,8,8,8,7,6,5,4,3,2,1,0,8,8,...)

Instead of underlining terms, we will put parentheses around the terms we want to emphasize:

a(1) = 8 produces parentheses around a(1 + 9 = 10):

8,7,6,5,4,3,2,1,0,(8),8,8,8,8,8,8,8,8,7,6,5,4,3,2,1,0,8,8,...

a(2) = 7 produces parentheses around a(2 + 8 = 10), which is now already done. Then,

a(3) = 6 produces parentheses around a(3 + 7 = 10), which is already done. Then,

a(4) = 5 produces parentheses around a(4 + 6 = 10) - already done. Then,

a(5) = 4 produces parentheses around a(5 + 5 = 10) - already done. Then,

a(6) = 3 produces parentheses around a(6 + 4 = 10) - already done. Then,

a(7) = 2 produces parentheses around a(7 + 3 = 10) - already done. Then,

a(8) = 1 produces parentheses around a(8 + 2 = 10) - already done. Then,

a(9) = 0 produces parentheses around a(9 + 1 = 10) - already done. Then,

a(10) = 8 produces parentheses around a(10 + 9 = 19):

8,7,6,5,4,3,2,1,0,(8),8,8,8,8,8,8,8,8,(7),6,5,4,3,2,1,0,8,8,...

a(11) = 8 produces parentheses around a(11 + 9 = 20):

8,7,6,5,4,3,2,1,0,(8),8,8,8,8,8,8,8,8,(7),(6),5,4,3,2,1,0,8,8,...

a(12) = 8 produces parentheses around a(12 + 9 = 21):

8,7,6,5,4,3,2,1,0,(8),8,8,8,8,8,8,8,8,(7),(6),(5),4,3,2,1,0,8,8,...

a(13) = 8 produces parentheses around a(13 + 9 = 22):

8,7,6,5,4,3,2,1,0,(8),8,8,8,8,8,8,8,8,(7),(6),(5),(4),3,2,1,0,8,8,...

a(14) = 8 produces parentheses around a(14 + 9 = 23):

8,7,6,5,4,3,2,1,0,(8),8,8,8,8,8,8,8,8,(7),(6),(5),(4),(3),2,1,0,8,8,...

a(15) = 8 produces parentheses around a(15 + 9 = 24):

8,7,6,5,4,3,2,1,0,(8),8,8,8,8,8,8,8,8,(7),(6),(5),(4),(3),(2),1,0,8,8,...

a(16) = 8 produces parentheses around a(16 + 9 = 25):

8,7,6,5,4,3,2,1,0,(8),8,8,8,8,8,8,8,8,(7),(6),(5),(4),(3),(2),(1),0,8,8,...

a(17) = 8 produces parentheses around a(17 + 9 = 26):

8,7,6,5,4,3,2,1,0,(8),8,8,8,8,8,8,8,8,(7),(6),(5),(4),(3),(2),(1),(0),8,8,...

a(18) = 7 produces parentheses around a(18 + 8 = 26) - already done. Then,

a(19) = 6 produces parentheses around a(19 + 7 = 26) - already done. Etc.

We see in this small example that the parenthesized terms reproduce the initial sequence:

(8),(7),(6),(5),(4),(3),(2),(1),(0),...

The same is true for the subsequence of non-parenthesized terms:

8, 7, 6, 5, 4, 3, 2, 1, 0, 8, 8, 8, 8, 8, 8, 8, 8,...

CROSSREFS

Cf. A307183 (first binary example of such fractal sequences), A307332 (ternary), A307333 (quaternary), A307335 (quinary), A307336 (senary), A307337 (septuary), A307338 (octal), A307340 (decimal).

Sequence in context: A022964 A023450 A141726 * A031310 A161813 A284809

Adjacent sequences:  A307336 A307337 A307338 * A307340 A307341 A307342

KEYWORD

base,nonn

AUTHOR

Eric Angelini and Carole Dubois, Apr 02 2019

STATUS

approved

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Last modified January 23 07:07 EST 2020. Contains 331168 sequences. (Running on oeis4.)