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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. 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, 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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified June 22 20:16 EDT 2021. Contains 345388 sequences. (Running on oeis4.)