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%I #10 Apr 04 2019 22:49:48
%S 5,4,3,2,1,0,5,5,5,5,5,5,4,3,2,1,0,5,5,4,3,2,1,5,5,0,5,5,5,5,5,4,4,3,
%T 2,1,0,5,3,2,1,5,4,5,5,0,5,3,5,2,1,0,5,5,5,5,5,5,5,5,5,5,4,4,3,2,1,0,
%U 5,4,4,3,2,1,3,2,0,5,1,5,5,3,2,4,1,5,0,5,5,5,4,3,5,2,1,0,5,5,5,0,5,5,5,5,5,3,5,5,5,5,4
%N A fractal senary (6 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.
%C This is defined to be the lexicographically earliest senary 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); if a(n) = 4, underline a(n+5); if a(n) = 5, underline a(n+6). 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="/A307336/b307336.txt">Table of n, a(n) for n = 1..2006</a>
%e The sequence starts (5,4,3,2,1,0,5,5,5,5,5,5,4,3,2,1,0,5,5,...)
%e Instead of underlining terms, we will put parentheses around the terms we want to emphasize:
%e a(1) = 5 produces parentheses around a(1 + 6 = 7):
%e 5,4,3,2,1,0,(5),5,5,5,5,5,4,3,2,1,0,5,5,...
%e a(2) = 4 produces parentheses around a(2 + 5 = 7), which is now already done. Then,
%e a(3) = 3 produces parentheses around a(3 + 4 = 7), which is already done. Then,
%e a(4) = 2 produces parentheses around a(4 + 3 = 7) - already done. Then,
%e a(5) = 1 produces parentheses around a(5 + 2 = 7) - already done. Then,
%e a(6) = 0 produces parentheses around a(6 + 1 = 7) - already done. Then,
%e a(7) = 5 produces parentheses around a(7 + 6 = 13):
%e 5,4,3,2,1,0,(5),5,5,5,5,5,(4),3,2,1,0,5,5,...
%e a(8) = 5 produces parentheses around a(8 + 6 = 14):
%e 5,4,3,2,1,0,(5),5,5,5,5,5,(4),(3),2,1,0,5,5,...
%e a(9) = 5 produces parentheses around a(9 + 6 = 15):
%e 5,4,3,2,1,0,(5),5,5,5,5,5,(4),(3),(2),1,0,5,5,...
%e a(10) = 5 produces parentheses around a(10 + 6 = 16):
%e 5,4,3,2,1,0,(5),5,5,5,5,5,(4),(3),(2),(1),0,5,5,...
%e a(11) = 5 produces parentheses around a(11 + 6 = 17):
%e 5,4,3,2,1,0,(5),5,5,5,5,5,(4),(3),(2),(1),(0),5,5,...
%e a(12) = 5 produces parentheses around a(12 + 6 = 18):
%e 5,4,3,2,1,0,(5),5,5,5,5,5,(4),(3),(2),(1),(0),(5),5,...
%e a(13) = 4 produces parentheses around a(13 + 5 = 18) - already done. Then,
%e a(14) = 3 produces parentheses around a(14 + 4 = 18) - already done. Then,
%e a(15) = 2 produces parentheses around a(15 + 3 = 18) - already done. Then,
%e a(16) = 1 produces parentheses around a(16 + 2 = 18) - already done. Then,
%e a(17) = 0 produces parentheses around a(17 + 1 = 18) - already done. Etc.
%e We see in this small example that the parenthesized terms reproduce the initial sequence:
%e (5),(4),(3),(2),(1),(0),(5)...
%e The same is true for the subsequence of non-parenthesized terms:
%e 5, 4, 3, 2, 1, 0, 5, 5, 5, 5, 5, 5,...
%Y Cf. A307183 (first binary example of such fractal sequences), A307332 (ternary), A307333 (quaternary), A307335 (quinary), A307337 (septuary), A307338 (octal), A307339 (nonary), A307340 (decimal).
%K base,nonn
%O 1,1
%A _Eric Angelini_ and _Carole Dubois_, Apr 02 2019
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