The binary "lizard sequence":
The field of numerical sequences is lasting for thousands of years. Eg: Prime numbers, the well-known “Fibonacci sequence” or the more recent "Thue-Morse" sequence.
In this domain, now & then, new and simple ideas are brought and lead to questions of great difficulty.
Since 2005 or so, the mathematician Eric Angelini proposes a new class of numerical sequences, very peculiar and fascinating : the "decimation-like sequences".
Such sequences deserve the wording of fractal sequences. Among them one binary sequence was named the binary "lizard sequence".
In the following binary sequence:
0 1 0 0 0 1 0 1 0 0 1 0 0 0 0 1 0 1 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 1 0 . .
when taking one digit over three (digit n°3, n°6, n°9 ..) you get a sequence actually identical to the initial sequence
0 1 0 0 0 1 0 1 0 0 1 0 . .
and – moreover - the digits left show again the same sequence :
0 1 0 0 0 1 0 1 0 0 1 0 0 0 0 1 0 1 0 0 0 0 1 1 . .
This is the only binary sequence starting with a zero and showing such an extreme fractal property !
No needs to "zoom down" to discover a smaller set identical to the whole !
A few open questions & possible projects for the mathematicians :
Computing the first 100,000 digits show 68,967 ‘0’ and 31,033 ‘1’.
- Will the ratio number of ‘0’ / ’1’ converge or not ? or is there a limit to this ratio ?
- Is the sequence recurring after a certain rank ? Likely not but how to prove it ?
- Does the sequence reveal chains of ‘0’ as long as one wishes or is there a maximum chain of consecutive ‘0’ ?
- Any chance for the sequence to be transcendental ?
Ref : Article by Jean-Paul Delahaye "Pour la Science" Mars 2007
Open challenges for artists :
Before A.I. machines (eg : iArtist) start doing it, play with the sequence likewise well-known artists did with Prime numbers, Fibonacci or Thue-Morse sequences.
This sequence is remarkable in many ways : its concept, its beauty and its timelessness. Translating it into sketches or drawings and paintings would reflect its unusual beauty.