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A331814
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Fourier coefficients of the boundary of the Mandelbrot set (normalized by a power of two).
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0
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-1, 1, -8, 15, 0, -94, -512, 987, 0, -7346, 65536, -122058, 0, -2757820, -22020096, 59250963, 0, -329425898, 2617245696, -4805611678, -34359738368, -19403249316, 498216206336, -36302282082, 0, 14136557849100, -71399536328704, -88183884706356
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OFFSET
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0,3
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COMMENTS
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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.
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LINKS
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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).
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FORMULA
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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.
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EXAMPLE
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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.
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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