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A341419
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a(0) = 1, a(1) = 1, a(2^(n-1)..2^n-1) = fwht(0..2^(n-2)). Here "fwht" is the fast Walsh-Hadamard transform with natural ordering and without multiplication of any factors.
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1
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1, 1, 2, 0, 4, 2, 0, -2, 8, 6, 8, -2, 0, -2, -8, -2, 16, 14, 24, -2, 32, 14, -8, -18, 0, -2, -8, -2, -32, -18, -8, 14, 32, 30, 56, -2, 96, 46, -8, -50, 128, 94, 120, -34, -32, -50, -136, -18, 0, -2, -8, -2, -32, -18, -8, 14, -128, -98, -136, 30, -32, 14, 120, 46, 64, 62
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OFFSET
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0,3
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COMMENTS
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This sequence is a rough integer-valued approximation to one of the nontrivial solutions to f(n) = a*fwht(f(n)).
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LINKS
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FORMULA
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a(2^n) = 2^n.
a(2^n + 1) = 2^n-2 for n > 0.
a(2^n + 2) = 8*(2^(n-2) - 1) = A159741(n-2) for n > 1.
a(2^n + 3) = -2 for n > 1.
a(2^n + 4) = 32*(2^(n-3) - 1) = A175165(n-3) for n > 2.
a(2^n + 5) = 2*(2^n - 9) for n > 2.
a(2^n + 6) = -8 for n > 2.
a(2^n + 7) = -2*(8 * 2^(n-3) - 7) for n > 2.
a(2^n + 8) = 64*(2^(n-3) - 2) for n > 3.
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PROG
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(MATLAB)
a(1) = 1;
a(2) = 1;
while length(a) < max_n
w = fwht(a, [], 'hadamard')*length(a);
%w = myfwht(a); % own implementation for documentation purpose
a = [a w];
end
end
function w = myfwht(in)
h = 1;
while h < length(in)
for i = 1:h*2:length(in)
for j = i:i+h-1
x = in(j);
y = in(j+h);
in(j) = x+y;
in(j+h) = x-y;
end
end
h = h*2;
end
w = in;
end
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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