

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..k1} 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
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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 6681. Reprinted in Donald E. Knuth, Selected Papers on Fun and Games, CSLI Publications, 2011, pages 571614.


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.


KEYWORD



AUTHOR



STATUS

approved



