OFFSET
0,3
COMMENTS
a(n) = 2^(2*n+1)*b(n) where b(n) is the unique sequence of complex numbers such that f(z) := z + Sum_{n>=0} (b(n)*z^-n) defines an analytic homeomorphism (biholomorphic bijection) between the complement of the unit disk and the complement of the Mandelbrot set, sometimes known as the "Jungreis function". (The b(n) are rationals, so we multiply them by the appropriate power of two to make them integers; this is equivalent to a simple rescaling of the complex plane.) It is conjectured that |b(n)| <= 1/n, so |a(n)| <= 2^(2*n+1)/n.
Note that the table given in Ewing and Schober (1992) gives the coefficients of the inverse series (contrary to what the text itself says): it's not wrong, it's just mislabeled.
LINKS
John H. Ewing and Glenn Schober, On the Coefficients of the Mapping to the Exterior of the Mandelbrot set, Michigan Math. J. 37 (1990), 315-320.
John H. Ewing & Glenn Schober, The area of the Mandelbrot set, Numer. Math. 61 (1992) 59-72 (note that table 1 gives the coefficients of the INVERSE series).
Irwin Jungreis, The uniformization of the complement of the Mandelbrot set, Duke Math. J. 4 (1985), 935-938.
David A. Madore, Sage code
FORMULA
a(m)=B(0,m+1) where B(0,0)=1/2 and by downwards induction on k we have B(k-1,m) = 2^(2^(k+1)-1)*B(k,m) - 2^(2^(k+1)-4)*Sum_{j=2^k-1..m-2^k+1} (B(k-1,j)*B(k-1,m-j) - 2*B(0,m-2^k+1)) if m >= 2^k-1, 0 otherwise.
EXAMPLE
a(0)=-1 because B(1,1)=0 and B(0,1) = 8*B(1,1) - 2*B(0,0) = -1; then a(1)=1 because B(1,2)=0 and B(0,2) = 8*B(1,2) - B(0,1)^2 - 2*B(0,1) = 1.
CROSSREFS
KEYWORD
sign
AUTHOR
David A. Madore, Jan 27 2020
STATUS
approved