A341419 - OEIS

%I #37 Mar 29 2021 12:56:15

%S 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,

%T -8,-2,-32,-18,-8,14,32,30,56,-2,96,46,-8,-50,128,94,120,-34,-32,-50,

%U -136,-18,0,-2,-8,-2,-32,-18,-8,14,-128,-98,-136,30,-32,14,120,46,64,62

%N 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.

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

%H Thomas Scheuerle, <a href="/A341419/b341419.txt">Table of n, a(n) for n = 0..16383</a>

%F a(2^n) = 2^n.

%F a(2^n + 1) = 2^n-2 for n > 0.

%F a(2^n + 2) = 8*(2^(n-2) - 1) = A159741(n-2) for n > 1.

%F a(2^n + 3) = -2 for n > 1.

%F a(2^n + 4) = 32*(2^(n-3) - 1) = A175165(n-3) for n > 2.

%F a(2^n + 5) = 2*(2^n - 9) for n > 2.

%F a(2^n + 6) = -8 for n > 2.

%F a(2^n + 7) = -2*(8 * 2^(n-3) - 7) for n > 2.

%F a(2^n + 8) = 64*(2^(n-3) - 2) for n > 3.

%o (MATLAB)

%o function a = A341419(max_n)

%o a(1) = 1;

%o a(2) = 1;

%o     while length(a) < max_n

%o         w = fwht(a,[],'hadamard')*length(a);

%o         %w = myfwht(a); % own implementation for documentation purpose

%o         a = [a w];

%o     end

%o end

%o function w = myfwht(in)

%o     h = 1;

%o     while h < length(in)

%o         for i = 1:h*2:length(in)

%o             for j = i:i+h-1

%o                 x = in(j);

%o                 y = in(j+h);

%o                 in(j) = x+y;

%o                 in(j+h) = x-y;

%o             end

%o         end

%o         h = h*2;

%o     end

%o     w = in;

%o end

%Y Cf. A159741, A175165.

%K sign,look

%O 0,3

%A _Thomas Scheuerle_, Mar 24 2021