A341419 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.

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

Offset: 0

Author

Thomas Scheuerle, Mar 24 2021

Keywords

• sign
• look

Comments

This sequence is a rough integer-valued approximation to one of the nontrivial solutions to f(n) = a*fwht(f(n)).

Links

• Thomas Scheuerle, Table of n, a(n) for n = 0..16383

Crossrefs

Cf. A159741, A175165.

Programs

MATLAB

function a = A341419(max_n)
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

Formula

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.