Hi
I'm an independent researcher exploring a potential link between the shortcut Collatz function (T(n)) and ECA Rule 54 (single cell evolution). I need help verifying a formula from the MathWorld page for Rule 54.
The page presents this piecewise formula: $$ a(n) = \begin{cases} \frac{1}{15}(4^{n+2} - 1) & \text{if } n \text{ is even} \newline \frac{7}{15}(4^{n+1} - 1) & \text{if } n \text{ is odd} \end{cases} \quad (n \ge 1) $$
https://mathworld.wolfram.com/Rule54.html My work proves that T(n) = wH(a(n)) - 1 (where wH is binary Hamming weight), assuming a(n) represents the normalized decimal value (odd-part) of the n'th row pattern.
Could someone please help clarify:
- Is the interpretation of
a(n) (as normalized value) correct for Rule 54 (single cell) ?
- What is the source or derivation for this specific
a(n) formula?
- Is the formula original to MathWorld?
- If a primary source isn't readily available, is citing the MathWorld page directly considered acceptable practice for this formula in a formal paper?
Verifying this is crucial for my paper connecting Collatz to Rule 54. While Dr. Weisstein might be the most direct source, perhaps he isn't very active here. Any insights, references, or verification from the community would be greatly appreciated.
Thanks so much!
Rudi B. Stranden
