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

Lars Blomberg, Table of n, a(n) for n = 1..1000

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

Adjacent sequences:  A307240 A307241 A307242 * A307244 A307245 A307246

KEYWORD

base,nonn,nice

AUTHOR

Eric Angelini and Lars Blomberg, Mar 30 2019

STATUS

approved

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Last modified August 7 12:13 EDT 2020. Contains 336276 sequences. (Running on oeis4.)