A307332 - OEIS

%I #12 Apr 04 2019 22:49:09

%S 2,1,0,2,2,2,1,0,2,2,1,2,2,0,1,0,2,2,2,2,1,2,2,1,0,1,2,0,2,2,2,2,2,1,

%T 2,2,0,1,0,1,1,2,0,2,2,0,2,2,2,2,1,2,2,2,0,1,0,1,2,1,2,2,1,0,2,2,2,2,

%U 0,2,2,1,2,2,0,1,2,2,1,2,0,0,1,2,1,0,2,2,2,1,2,2,2,1,0,2,2,2,2,2,2,0,2,2,1,1,2,2,2,0,1,2

%N A fractal ternary 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.

%C This is defined to be the lexicographically earliest ternary sequence with the following property:

%C If a(n) = 0, underline a(n+1); if a(n) = 1, underline a(n+2) ; if a(n) = 2, underline a(n+3). 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).

%H Carole Dubois, <a href="/A307332/b307332.txt">Table of n, a(n) for n = 1..2001</a>

%e The sequence starts (2, 1, 0, 2, 2, 2, 1, 0, 2, 2, 1, 2, 2, 0, 1, 0,...)

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

%e a(1) = 2 produces parentheses around a(1 + 3 = 4):

%e 2, 1, 0, (2), 2, 2, 1, 0, 2, 2, 1, 2, 2, 0, 1, 0,...

%e a(2) = 1 produces parentheses around a(2 + 2 = 4), which is now already done. Then,

%e a(3) = 0 produces parentheses around a(3 + 1 = 4), which is already done. Then,

%e a(4) = 2 produces parentheses around a(4 + 3 = 7):

%e 2, 1, 0, (2), 2, 2, (1), 0, 2, 2, 1, 2, 2, 0, 1, 0,...

%e a(5) = 2 produces parentheses around a(5 + 3 = 8):

%e 2, 1, 0, (2), 2, 2, (1), (0), 2, 2, 1, 2, 2, 0, 1, 0,...

%e a(6) = 2 produces parentheses around a(6 + 3 = 9):

%e 2, 1, 0, (2), 2, 2, (1), (0), (2), 2, 1, 2, 2, 0, 1, 0,...

%e a(7) = 1 produces parentheses around a(7 + 2 = 9), which is already done. Then,

%e a(8) = 0 produces parentheses around a(8 + 1 = 9) - already done. Then,

%e a(9) = 2 produces parentheses around a(9 + 3 = 12):

%e 2, 1, 0, (2), 2, 2, (1), (0), (2), 2, 1, (2), 2, 0, 1, 0,...

%e a(10) = 2 produces parentheses around a(10 + 3 = 13):

%e 2, 1, 0, (2), 2, 2, (1), (0), (2), 2, 1, (2), (2), 0, 1, 0,...

%e a(11) = 1 produces parentheses around a(11 + 2 = 13) - already done. Then,

%e a(12) = 2 produces parentheses around a(12 + 3 = 15):

%e 2, 1, 0, (2), 2, 2, (1), (0), (2), 2, 1, (2), (2), 0, (1), 0,...

%e a(13) = 2 produces parentheses around a(13 + 3 = 16):

%e 2, 1, 0, (2), 2, 2, (1), (0), (2), 2, 1, (2), (2), 0, (1), (0),... and so on.

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

%e (2),(1),(0),(2),(2),(2),(1),(0),...

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

%e 2, 1, 0, 2, 2, 2, 1, 0,...

%Y Cf. A307183 (first binary example of such fractal sequences), A307333 (quaternary), A307335 (quinary), A307336 (senary), A307337 (septuary), A307338 (octal), A307339 (nonary), A307340 (decimal).

%K base,nonn

%O 1,1

%A _Eric Angelini_ and _Carole Dubois_, Apr 02 2019