A332252 -id:A332252 - cp's OEIS frontend

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.

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

Offset: 0

Rémy Sigrist, Feb 08 2020

The representation of {f(n)} corresponds to a Lévy C Curve.

			The first terms, alongside f(n) and A000120(n), are:
  n   a(n)  f(n)    A000120(n)
  --  ----  ------  ----------
   0     0       0           0
   1     1       1           1
   2     1     1+i           1
   3     1   1+2*i           2
   4     0     2*i           1
   5     0     3*i           2
   6    -1  -1+3*i           2
   7    -2  -2+3*i           3
   8    -2  -2+2*i           1
   9    -2  -2+3*i           2
  10    -3  -3+3*i           2
  11    -4  -4+3*i           3
  12    -4  -4+2*i           2
  13    -5  -5+2*i           3
  14    -5    -5+i           3
  15    -5      -5           4
  16    -4      -4           1
From _Kevin Ryde_, Sep 24 2020: (Start)
  n    =   2^9 +   2^8 +     2^5 +     2^2 +     2^1 = 806
  f(n) = 1*b^9 + i*b^8 + i^2*b^5 + i^3*b^2 + i^4*b^1 = 23 + 37*i
so a(806) = 23 and A332252(806) = 37.
(End)

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Michael Beeler, R. William Gosper, and Richard Schroeppel, HAKMEM, MIT Artificial Intelligence Laboratory report AIM-239, February 1972. Item 135, The "C" Curve, by Gosper, page 65. Also HTML transcription.
Robert Ferréol (MathCurve), Courbe du C (ou courbe de Lévy) [in French]
Paul Lévy, Les courbes planes ou gauches et les surfaces composées de parties semblables au tout, Journal de l'École Polytechnique, July 1938 pages 227-247, and continued October 1938 pages 249-292.
Kevin Ryde, Iterations of the Lévy C Curve, section Coordinates.
Rémy Sigrist, Colored representation of f(n) in the complex plane for n = 0..2^20 (where the hue is function of n)
Rémy Sigrist, Representation of f(n) for n=0..32 in the complex plan
Wikipedia, Lévy C Curve
Index entries for sequences related to coordinates of 2D curves

Cf. A332252 (imaginary part), A000120 (segment direction), A179868 (segment direction mod 4).

Cf. A332383 (dragon curve).

{ z=0; for (n=0, 67, print1 (real(z) ", "); z += I^hammingweight(n)) }

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

For any k >= 0:
- a(2^(4*k))   =    (-4)^k,
- a(2^(4*k+1)) =    (-4)^k,
- a(2^(4*k+2)) =         0,
- a(2^(4*k+3)) = -2*(-4)^k.
From Kevin Ryde, Sep 24 2020: (Start)
With complex b = 1+i,
f(2*n) = b*f(n) and f(2*n+1) = f(2*n) + i^A000120(2*n), expand and step.
f(2^k + r) = b^k + i*f(r), for 0 <= r < 2^k, replication.
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.
(End)

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.

Original entry on oeis.org

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

Offset: 0

Rémy Sigrist, Oct 29 2021

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).

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))

See A332251 for a similar sequence.

Cf. A000120, A065620, A146559, A348691, A332251, A332252.

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) }

a(2^k) = A146559(k) for any k >= 0.

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.

Original entry on oeis.org

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

Rémy Sigrist, Oct 29 2021

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).

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))

See A332251, A332252 for a similar sequence.

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

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) }

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

cp's OEIS Frontend

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.

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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.

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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.

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