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A332251 a(n) is the real part of f(n) defined by f(0) = 0 and f(n+1) = f(n) + i^A000120(n) (where i denotes the imaginary unit). Sequence A332252 gives imaginary parts.

%I A332251 #34 Sep 25 2020 04:14:17
%S A332251 0,1,1,1,0,0,-1,-2,-2,-2,-3,-4,-4,-5,-5,-5,-4,-4,-5,-6,-6,-7,-7,-7,-6,
%T A332251 -7,-7,-7,-6,-6,-5,-4,-4,-4,-5,-6,-6,-7,-7,-7,-6,-7,-7,-7,-6,-6,-5,-4,
%U A332251 -4,-5,-5,-5,-4,-4,-3,-2,-2,-2,-1,0,0,1,1,1,0,0,-1,-2
%N A332251 a(n) is the real part of f(n) defined by f(0) = 0 and f(n+1) = f(n) + i^A000120(n) (where i denotes the imaginary unit). Sequence A332252 gives imaginary parts.
%C A332251 The representation of {f(n)} corresponds to a Lévy C Curve.
%H A332251 Rémy Sigrist, <a href="/A332251/b332251.txt">Table of n, a(n) for n = 0..8192</a>
%H A332251 Michael Beeler, R. William Gosper, and Richard Schroeppel, <a href="https://dspace.mit.edu/handle/1721.1/6086">HAKMEM</a>, MIT Artificial Intelligence Laboratory report AIM-239, February 1972.  Item 135, The "C" Curve, by Gosper, page 65.  Also <a href="http://www.inwap.com/pdp10/hbaker/hakmem/flows.html#item135">HTML transcription</a>.
%H A332251 Robert Ferréol (MathCurve), <a href="https://www.mathcurve.com/fractals/c/c.shtml">Courbe du C (ou courbe de Lévy)</a> [in French]
%H A332251 Paul Lévy, <a href="https://gallica.bnf.fr/ark:/12148/bpt6k57344323/f53.image">Les courbes planes ou gauches et les surfaces composées de parties semblables au tout</a>, Journal de l'École Polytechnique, July 1938 pages 227-247, and <a href="https://gallica.bnf.fr/ark:/12148/bpt6k57344820/f3.image">continued</a> October 1938 pages 249-292.
%H A332251 Kevin Ryde, <a href="http://user42.tuxfamily.org/c-curve/index.html">Iterations of the Lévy C Curve</a>, section Coordinates.
%H A332251 Rémy Sigrist, <a href="/A332251/a332251.png">Colored representation of f(n) in the complex plane for n = 0..2^20</a> (where the hue is function of n)
%H A332251 Rémy Sigrist, <a href="/A332251/a332251_1.png">Representation of f(n) for n=0..32 in the complex plan</a>
%H A332251 Wikipedia, <a href="https://en.wikipedia.org/wiki/L%C3%A9vy_C_curve">Lévy C Curve</a>
%H A332251 <a href="/index/Con#coordinates_2D_curves">Index entries for sequences related to coordinates of 2D curves</a>
%F A332251 For any k >= 0:
%F A332251 - a(2^(4*k))   =    (-4)^k,
%F A332251 - a(2^(4*k+1)) =    (-4)^k,
%F A332251 - a(2^(4*k+2)) =         0,
%F A332251 - a(2^(4*k+3)) = -2*(-4)^k.
%F A332251 From _Kevin Ryde_, Sep 24 2020: (Start)
%F A332251 With complex b = 1+i,
%F A332251 f(2*n) = b*f(n) and f(2*n+1) = f(2*n) + i^A000120(2*n), expand and step.
%F A332251 f(2^k + r) = b^k + i*f(r), for 0 <= r < 2^k, replication.
%F A332251 f(n) = Sum_{j=0..t} i^j*b^k[j] where binary n = 2^k[0] + ... + 2^k[t] with descending powers k[0] > ... > k[t] >= 0, so change binary to base b with rotating coefficient i^0, i^1, i^2, ... at each 1-bit.
%F A332251 (End)
%e A332251 The first terms, alongside f(n) and A000120(n), are:
%e A332251   n   a(n)  f(n)    A000120(n)
%e A332251   --  ----  ------  ----------
%e A332251    0     0       0           0
%e A332251    1     1       1           1
%e A332251    2     1     1+i           1
%e A332251    3     1   1+2*i           2
%e A332251    4     0     2*i           1
%e A332251    5     0     3*i           2
%e A332251    6    -1  -1+3*i           2
%e A332251    7    -2  -2+3*i           3
%e A332251    8    -2  -2+2*i           1
%e A332251    9    -2  -2+3*i           2
%e A332251   10    -3  -3+3*i           2
%e A332251   11    -4  -4+3*i           3
%e A332251   12    -4  -4+2*i           2
%e A332251   13    -5  -5+2*i           3
%e A332251   14    -5    -5+i           3
%e A332251   15    -5      -5           4
%e A332251   16    -4      -4           1
%e A332251 From _Kevin Ryde_, Sep 24 2020: (Start)
%e A332251   n    =   2^9 +   2^8 +     2^5 +     2^2 +     2^1 = 806
%e A332251   f(n) = 1*b^9 + i*b^8 + i^2*b^5 + i^3*b^2 + i^4*b^1 = 23 + 37*i
%e A332251 so a(806) = 23 and A332252(806) = 37.
%e A332251 (End)
%o A332251 (PARI) { z=0; for (n=0, 67, print1 (real(z) ", "); z += I^hammingweight(n)) }
%o A332251 (PARI) a(n) = my(v=binary(n),s=1); for(i=2,#v, if(v[i],v[i]=(s*=I))); real(subst(Pol(v),'x,1+I)); \\ _Kevin Ryde_, Sep 24 2020
%Y A332251 Cf. A332252 (imaginary part), A000120 (segment direction), A179868 (segment direction mod 4).
%Y A332251 Cf. A332383 (dragon curve).
%K A332251 sign,look,base
%O A332251 0,8
%A A332251 _Rémy Sigrist_, Feb 08 2020