A307333 - OEIS

%I #11 Apr 04 2019 22:49:26

%S 3,2,1,0,3,3,3,3,2,1,0,3,3,2,1,3,3,0,1,3,2,3,3,0,2,3,3,1,2,3,3,0,2,1,

%T 3,3,1,3,3,0,3,2,3,3,1,0,2,3,1,3,2,0,3,1,3,1,3,3,3,2,0,3,2,3,3,3,0,3,

%U 1,0,3,2,2,3,1,3,0,2,3,3,1,1,3,1,2,3,3,3,3,2,0,3,2,3,0,3,2,3,1,3,3,0,3,1,0,2,2,3,3,1,3,3,0

%N A fractal quaternary 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 quaternary 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); if a(n) = 3, underline a(n+4). 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="/A307333/b307333.txt">Table of n, a(n) for n = 1..2004</a>

%e The sequence starts (3, 2, 1, 0, 3, 3, 3, 3, 2, 1, 0, 3, 3, 2, 1, 3, 3, 0,...)

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

%e a(1) = 3 produces parentheses around a(1 + 4 = 5):

%e 3, 2, 1, 0, (3,) 3, 3, 3, 2, 1, 0, 3, 3, 2, 1, 3, 3, 0,...

%e a(2) = 2 produces parentheses around a(2 + 3 = 5), which is now already done. Then,

%e a(3) = 1 produces parentheses around a(3 + 2 = 5), which is already done. Then,

%e a(4) = 0 produces parentheses around a(4 + 1 = 5), which is already done. Now,

%e a(5) = 3 produces parentheses around a(5 + 4 = 9):

%e 3, 2, 1, 0, (3,) 3, 3, 3, (2), 1, 0, 3, 3, 2, 1, 3, 3, 0,...

%e a(6) = 3 produces parentheses around a(6 + 4 = 10):

%e 3, 2, 1, 0, (3,) 3, 3, 3, (2), (1), 0, 3, 3, 2, 1, 3, 3, 0,...

%e a(7) = 3 produces parentheses around a(7 + 4 = 11):

%e 3, 2, 1, 0, (3,) 3, 3, 3, (2), (1), (0), 3, 3, 2, 1, 3, 3, 0,...

%e a(8) = 3 produces parentheses around a(8 + 4 = 12):

%e 3, 2, 1, 0, (3,) 3, 3, 3, (2), (1), (0), (3), 3, 2, 1, 3, 3, 0,...

%e a(9) = 2 produces parentheses around a(9 + 3 = 12) - already done. Then,

%e a(10) = 1 produces parentheses around a(10 + 2 = 12) - already done. Then,

%e a(11) = 0 produces parentheses around a(11 + 1 = 12) - already done. Then,

%e a(12) = 3 produces parentheses around a(12 + 4 = 16):

%e 3, 2, 1, 0, (3,) 3, 3, 3, (2), (1), (0), (3), 3, 2, 1, (3), 3, 0,...

%e a(13) = 3 produces parentheses around a(13 + 4 = 17):

%e 3, 2, 1, 0, (3,) 3, 3, 3, (2), (1), (0), (3), 3, 2, 1, (3), (3), 0,...

%e a(14) = 2 produces parentheses around a(14 + 3 = 17) - already done. Then,

%e a(15) = 1 produces parentheses around a(15 + 2 = 17) - already done. Etc.

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

%e (3),(2),(1),(0),(3),(3),(3),...

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

%e 3, 2, 1, 0, 3, 3, 3, 3, 2, 1, 0,...

%Y Cf. A307183 (first binary example of such fractal sequences), A307332 (ternary), 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