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A348690 For any nonnegative number n with binary expansion Sum_{k >= 0} b_k * 2^k, a(n) is the real 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 A348691 gives the imaginary part. 3
0, 1, 1, 0, 0, -1, -1, 0, -2, -1, -1, 2, -2, 1, 1, 2, -4, 1, 1, 4, 0, 3, 3, 0, -2, 3, 3, 2, 2, 1, 1, -2, -4, 5, 5, 4, 4, 3, 3, -4, 2, 3, 3, -2, 2, -3, -3, -2, 0, 5, 5, 0, 4, -1, -1, -4, 2, -1, -1, -2, -2, -3, -3, 2, 0, 9, 9, 0, 8, -1, -1, -8, 6, -1, -1, -6, -2 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,9
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
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 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) = A146559(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++); real(v) }
CROSSREFS
See A332251 for a similar sequence.
Sequence in context: A305749 A320748 A320747 * A238392 A144464 A138015
KEYWORD
sign,look,base
AUTHOR
Rémy Sigrist, Oct 29 2021
STATUS
approved

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Last modified May 21 15:47 EDT 2024. Contains 372738 sequences. (Running on oeis4.)