A348691 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} b_k * (i^Sum_{j = 0..k-1} b_j) * (1+i)^k (where i denotes the imaginary unit); sequence A348690 gives the real part.

0, 0, 1, 1, 2, 0, 1, -1, 2, -2, -1, -1, 0, -2, -1, 1, 0, -4, -3, 1, -2, 0, 1, 3, -2, -2, -1, 3, 0, 2, 3, 1, -4, -4, -3, 5, -2, 4, 5, 3, -2, 2, 3, 3, 4, 2, 3, -3, -4, 0, 1, 5, 2, 4, 5, -1, 2, 2, 3, -1, 4, -2, -1, -3, -8, 0, 1, 9, 2, 8, 9, -1, 2, 6, 7, -1, 8, -2

Offset: 0

Author

Rémy Sigrist, Oct 29 2021

Keywords

• sign
• look
• base

Comments

The function f defines a bijection from the nonnegative integers to the Gaussian integers.

The function f has similarities with A065620; here the nonzero digits in base 1+i cycle through powers of i, there nonzero digits in base 2 cycle through powers of -1.

If we replace 1's in binary expansions by powers of i from left to right (rather than right to left as here), then we obtain the Lévy C curve (A332251, A332252).

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..8191
• Chandler Davis and Donald Knuth, Number representations and Dragon Curves I, Journal of Recreational Mathematics, volume 3, number 2 (April 1970), pages 66-81. Reprinted in Donald E. Knuth, Selected Papers on Fun and Games, CSLI Publications, 2011, pages 571-614.
• Chandler Davis and Donald E. Knuth, Number Representations and Dragon Curves, Journal of Recreational Mathematics, volume 3, number 2, April 1970, pages 66-81, and number 3, July 1970, pages 133-149. [Cached copy, with permission]
• Rémy Sigrist, Colored representation of f(n) for n < 2^18 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 4)
• 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))

Crossrefs

See A332251, A332252 for a similar sequence.

Cf. A000120, A009545, A065620, A070939, A348690.

Programs

PARI

a(n) = { my (v=0, o=0, x); while (n, n-=2^x=valuation(n, 2); v+=I^o * (1+I)^x; o++); imag(v) }

Formula

a(2^k) = A009545(k) for any k >= 0.