A332204 a(n) is the real part of f(n) defined by f(0) = 0, and f(n+1) = f(n) + g((1+i)^(A065359(n) mod 8)) (where g(z) = z/gcd(Re(z), Im(z)) and i denotes the imaginary unit).

0, 1, 2, 3, 4, 5, 5, 6, 7, 8, 9, 9, 10, 11, 12, 13, 14, 15, 15, 16, 17, 17, 16, 17, 17, 18, 19, 20, 21, 22, 22, 23, 24, 25, 26, 26, 27, 28, 29, 30, 31, 31, 32, 31, 31, 32, 33, 33, 34, 35, 36, 37, 38, 39, 39, 40, 41, 42, 43, 43, 44, 45, 46, 47, 48, 49, 49, 50

Offset

0,3

Comments

The representation of {f(n)} resembles a Koch curve (see illustrations in Links section).

The sequence A065359 mod 8 gives the direction at each step as follows:

3 _ 2 _ 1

\_ | _/

\_ | _/

\|/

4 ------.------ 0

_/|\_

_/ | \_

_/ | \_

5 6 7

We can also build {f(n)} with A096268 as follows:

- start at the origin looking to the right,

- for k=0, 1, ...:

- move forward to the next lattice point

(this point is at distance 1 or sqrt(2)),

- if A096268(k)=0

then turn 45 degrees to the left

otherwise turn 90 degrees to the right,

- this connects the first differences of A065359 and A096268.

Links

Rémy Sigrist, Table of n, a(n) for n = 0..16384

Larry Riddle, Koch Curve

Rémy Sigrist, Illustration of first terms

Rémy Sigrist, Representation of f(n) in the complex plan for n = 0..2^14

Rémy Sigrist, PARI program for A332204

Index entries for sequences related to coordinates of 2D curves

Formula

a(2^k) = A217730(k) for any k >= 0.

a(4^k+m) + a(m) = A217730(2*k) for any k >= 0 and m = 0..4^k.

Example

The first terms, alongside f(n) and A065359(n), are:
  n   a(n)  f(n)   A065359(n)
  --  ----  -----  ----------
   0     0      0           0
   1     1      1           1
   2     2    2+i          -1
   3     3      3           0
   4     4      4           1
   5     5    5+i           2
   6     5  5+2*i           0
   7     6  6+2*i           1
   8     7  7+3*i          -1
   9     8  8+2*i           0
  10     9  9+2*i          -2
  11     9    9+i          -1
  12    10     10           0
  13    11     11           1
  14    12   12+i          -1
  15    13     13           0
  16    14     14           1

Mathematica

A065359[0] = 0;
A065359[n_] := -Total[(-1)^PositionIndex[Reverse[IntegerDigits[n, 2]]][1]];
g[z_] := z/GCD[Re[z], Im[z]];
Module[{n = 0}, Re[NestList[# + g[(1+I)^A065359[n++]] &, 0, 100]]] (* Paolo Xausa, Aug 28 2024 *)

Prog

(PARI) \\ See Links section.

Crossrefs

Cf. A065359, A096268, A217730, A332205 (imaginary part), A332206 (where f is real).

Sequence in context: A352241 A195181 A003005 * A245321 A006163 A331268

Adjacent sequences: A332201 A332202 A332203 * A332205 A332206 A332207

Keyword

nonn,base

Author

Rémy Sigrist, Feb 07 2020

Status

approved