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A307243
An irregular fractal binary sequence embedding three copies of itself. See comments for precise definition.
1
1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0
OFFSET
1
COMMENTS
The sequence has the following property: if a(n) = 0, underline a(n+1); if a(n) = 1, underline a(n+2). The terms underlined twice are the original sequence (copy #1), the terms underlined once are also (copy #2) and the non-underlined also (copy #3).
From Lars Blomberg: (Start)
I think this idea can be generalized:
Select some k >= 1, for example 2.
Initialize S = (k)(k-1)...0, for example S=210, and evaluate the underlining: the underlined amount is shown after '.', S=210.3
Initialize the "fetch pointers", f0=k, f1...fk=0 (using 0-based indexing);
Let p be the pointer to the next term to be evaluated, start with p=k=2;
The term indexed by p is shown in [], and the values of f0...fk are shown in ();
So for 210.[3]-(200), the underlined count is 3 - we should take the value from f2 which is 2, and calculate its underlining:
2102.[0]01-(201). Next is 0 so we take from f0:
21020.[1]1-(301) and so on.
Note that before «.» are the terms, and after «.» are the underlined, so a single array will suffice to store everything.
For k=2, I get:
2 1 0 2 2 2 2 1 0 1 2 2 2 2 2 2 1 0 0 1 1 2 2 0 2 2 1 2 1 2 0 2 2 2 1 0 2 2 2 0 2 1 2 2 2 1 2 2 1 2 2 0 1 0 0 2 0 2 1 1 1 2 1 1 2 2 2 0 2 2 2 2 2 1 0 2 0 2 2 2 2 0 2 2 2 1 2 2 1 2 2 2 2 1 2 2 1 2 1 2 0 1 2 2 0 0 1 2 0 1 0 1 2 0 2 0 2 1 2 2 1 1 2 1 0 2 2 2 2 1 1 0 2 2 0 2 2 2 2 2 0 1 2 2 1 2 2 2 2 2 1 0 2 2 2 2 0 1 2 2 1 2 2 2 2
And for k=3:
3 2 1 0 3 3 3 3 3 2 1 0 2 3 3 3 2 3 3 3 1 3 2 3 0 3 2 2 3 1 1 0 2 3 3 2 2 3 0 3 3 3 3 3 2 3 3 3 3 3 1 3 3 2 1 3 1 3 0 3 2 3 3 3 0 2 2 2 3 1 1 0 0 2 3 2 3 2 1 3 3 1 1 0 3 0 3 2 2 3 2 2 3 3 3 2 0 3 3 2 3 3 2 0 3 3 3 3 3 3 3 3 2 3 3 3 3 3 3 3 1 3 1 3 2 1 3 3 3 2 3 1 3 3 0 3 2 3 3 2 3 3 3 3 3 0 0 3 2 3 3 2 1 2 3 3 2 3 3 2 2 1 3 1 1
(End)
LINKS
EXAMPLE
The sequence starts with 1,0,1,1,1,0,0,1,1,1,1,0,1,1,0,1,...
Instead of underlining them, we will put () around the terms we want to emphasize.
a(1) = 1 produces parentheses around a(3):
1,0,(1),1,1,0,0,1,1,1,1,0,1,1,0,1,...
a(2) = 0 produces parentheses around a(3) again:
1,0,((1)),1,1,0,0,1,1,1,1,0,1,1,0,1,...
a(3) = 1 produces parentheses around a(5):
1,0,((1)),1,(1),0,0,1,1,1,1,0,1,1,0,1,...
a(4) = 1 produces parentheses around a(6):
1,0,((1)),1,(1),(0),0,1,1,1,1,0,1,1,0,1,...
a(5) = 1 produces parentheses around a(7):
1,0,((1)),1,(1),(0),(0),1,1,1,1,0,1,1,0,1,...
a(6) = 0 produces parentheses around a(7) again:
1,0,((1)),1,(1),(0),((0)),1,1,1,1,0,1,1,0,1,...
a(7) = 0 produces parentheses around a(8):
1,0,((1)),1,(1),(0),((0)),(1),1,1,1,0,1,1,0,1,...
a(8) = 1 produces parentheses around a(10):
1,0,((1)),1,(1),(0),((0)),(1),1,(1),1,0,1,1,0,1,...
a(9) = 1 produces parentheses around a(11):
1,0,((1)),1,(1),(0),((0)),(1),1,(1),(1),0,1,1,0,1,...
a(10) = 1 produces parentheses around a(12):
1,0,((1)),1,(1),(0),((0)),(1),1,(1),(1),(0),1,1,0,1,...
a(11) = 1 produces parentheses around a(13):
1,0,((1)),1,(1),(0),((0)),(1),1,(1),(1),(0),(1),1,0,1,...
a(12) = 0 produces parentheses around a(13) again:
1,0,((1)),1,(1),(0),((0)),(1),1,(1),(1),(0),((1)),1,0,1,...
a(13) = 1 produces parentheses around a(15):
1,0,((1)),1,(1),(0),((0)),(1),1,(1),(1),(0),((1)),(1),(0),1,...
a(14) = 1 produces parentheses around a(16):
1,0,((1)),1,(1),(0),((0)),(1),1,(1),(1),(0),((1)),(1),(0),(1),...
a(15) = 0 produces parentheses around a(16) again:
1,0,((1)),1,(1),(0),((0)),(1),1,(1),(1),(0),((1)),(1),(0),((1)),...
Etc.
We see in this small example that the doubly parenthesized terms of the last line slowly reconstruct the starting sequence:
((1)), ((0)), ((1)), ((1)), ...
The same holds for the singly parenthesized terms:
(1), (0), (1), (1), (1), (0), (1), (0), ...
And again by the non-parenthesized terms:
1, 0, 1, 1, ...
CROSSREFS
Cf. A307183 (no distinction between underlined once or twice: two copies only of the starting sequence).
Sequence in context: A099076 A282339 A175479 * A120530 A078616 A267800
KEYWORD
base,nonn,nice
AUTHOR
Eric Angelini and Lars Blomberg, Mar 30 2019
STATUS
approved