Sequence A006874: Number of Contenental Hyperbolic Components of Period N in the Continent of the Mandelbrot Set
This sequence, Sloane's A006874, appears in my Mu-Ency page Enumeration of Features, which gives more information. The sequence is represented as a function with the name Nc, in which the "c" stands for continental, and that word originates with Benoit B. Mandelbrot (see Continent). The values of the sequence are:
A006874: 1, 1, 2, 3, 4, 6, 6, 9, 10, 12, 10, 22, 12, 18, 24, 27, 16, 38, 18, 44, 36, 30, 22, 78, 36, 36, 50, 66, 28, 104, 30, 81, 60, 48, 72, 158, 36, 54, 72, 156, 40, 156, 42, 110, 152, 66, 46, 270, 78, 140, 96, 132, 52, 230, 120, 234, 108, 84, 58, 456, 60, 90, 228, 243, 144, 260, ...
We can calculate all the values in the sequence by doing this:
Nc(1) = 1 (special case)
Nc(n) = Σ(f:n≡0 mod f) [Nc(f) × Φ(n/f)]
The bit that says "f:n≡0 mod f" means "for all f such that n is divisible by f". Fancier mathematicians sometimes write this as "f|n" which is pronounced "f divides n".
The function Φ(n), called the "Euler totient function", counts irreducible simple fractions with a given denominator. For example Φ(14) is 6 because there are six fractions between 0 and 1 that have 14 in the denominator and cannot be reduced to a smaller denominator:
1/14, 3/14, 5/14, 9/14, 11/14, 13/14
The whole expression above, starting with Σ, is a sum of terms each of which is some value of the Nc() function multiplied by a value of the Φ() function. It can be explained more easily with an example. Considering 14 again, we first show all the ways 14 can be factored into positive integer parts, not counting "1" as a part, and allowing reordering. There are three:
14 = (14) = (7)(2) = (2)(7)
Then we put Φ in front of each parenthesised number, and multiply and add:
Nc(14) = Φ(14) + Φ(7)Φ(2) + Φ(2)Φ(7)
= 6 + 6×1 + 1×6
= 18
As this function applies to the Mandelbrot set specifically, the presence of the totient function Φ(n) comes from the Farey tree which produces all irreducible fractions and no others; and the multiplications for each factorization result from period scaling that is a special property of the iteration formula that produces Julia and Mandelbrot fractals.
Relationship to Fibonacci Partitions
The "Fibonacci partitions" are sequence A0119, which is the number of ways to express a number n as a sum of distinct Fibonacci numbers. For example:
n=3: 3 = 2+1 (2 ways): A0119(3)=2.
n=8: 8 = 5+3 = 5+2+1 (3 ways): A0119(8)=3.
n=16: 16 = 13+3 = 13+2+1 = 8+5+2+1 (4 ways): A0119(16)=4.
A paper by Felix Weinstein 1 discusses them in some detail and at one point comes to a derivation of a sequence that requires the same summation as above, leading to the same sequence. See also my pages on the Weinstein sequences A7896 and A7898.
History
I worked out this sequence and a few related ones in the late 1980s, and wrote the enumeration of features article as part of my ambitiously named article "The Mandelbrot Set — A Mathematical Description" that I shared with a few people in the late 1980s. I sent the sequences (this one, as well as A6875 and A6876) to N.J.A. Sloane in 1994 and they appeared in his book with Simon Plouffe 2. I published most of my article as web pages on individual Mandelbrot set topics a few years later via Earthlink.
1 : Felix V. Weinstein, "Notes on Fibonacci partitions." Experimental Mathematics 25(4 (2016): pp. 482-499.
2 : N.J.A. Sloane and Simon Plouffe, "The Encyclopedia of Integer Sequences", Academic Press (1995).
This page was written in the "embarrassingly readable" markup language RHTF, and was last updated on 2024 Mar 19.
s.30