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 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. 3
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 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. 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)) FORMULA a(2^k) = A009545(k) for any k >= 0. PROG (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) } CROSSREFS See A332251, A332252 for a similar sequence. Cf. A000120, A009545, A065620, A070939, A348690. Sequence in context: A281388 A127440 A118198 * A290885 A059881 A096568 Adjacent sequences: A348688 A348689 A348690 * A348692 A348693 A348694 KEYWORD sign,look,base AUTHOR Rémy Sigrist, Oct 29 2021 STATUS approved

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Last modified May 18 12:18 EDT 2024. Contains 372630 sequences. (Running on oeis4.)