A348761 For any nonnegative number n with binary expansion Sum_{k >= 0} b_k * 2^k, a(n) is the imaginary part of f(n) = Sum_{k >= 0} ((-1)^Sum_{j = 0..k-1} b_j) * (1+i)^k (where i denotes the imaginary unit); sequence A348760 gives the real part.
0, 0, 1, -1, 2, -2, -1, 1, 2, -2, -1, 1, 0, 0, 1, -1, 0, 0, 1, -1, 2, -2, -1, 1, 2, -2, -1, 1, 0, 0, 1, -1, -4, 4, 5, -5, 6, -6, -5, 5, 6, -6, -5, 5, -4, 4, 5, -5, 4, -4, -3, 3, -2, 2, 3, -3, -2, 2, 3, -3, 4, -4, -3, 3, -8, 8, 9, -9, 10, -10, -9, 9, 10, -10
Offset: 0
Links
- Rémy Sigrist, Table of n, a(n) for n = 0..8191
- Rémy Sigrist, Colored representation of f(n) for n < 2^20 in the complex plane (the hue is function of n)
- Rémy Sigrist, Colored representation of f(n) for n < 2^18 in the complex plane (the color is function of A000120(n) mod 2)
- Rémy Sigrist, Colored representation of f(n) for n < 2^18 in the complex plane (the color is function of the binary length of n, A070939(n))
- Rémy Sigrist, Colored representation of f(n) for n < 2^18 in the complex plane (the color is function of A000120(n), darker shades correspond to higher values)
Programs
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PARI
a(n) = { my (v=0, k, o=-1); while (n, n-=2^k=valuation(n,2); v+=(1+I)^k * (-1)^o++); imag(v) }
Formula
a(2^k) = A009545(k) for any k >= 0.
Comments