A332206 -id:A332206 - cp's OEIS frontend

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

Rémy Sigrist, Feb 07 2020

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.

			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

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

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

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 *)

PARI
```
\\ See Links section.
```

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.

A332205 a(n) is the imaginary 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).

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 2, 2, 3, 2, 2, 1, 0, 0, 1, 0, 0, 1, 2, 2, 3, 4, 5, 6, 7, 7, 8, 7, 7, 8, 9, 9, 10, 9, 9, 8, 7, 7, 8, 7, 7, 6, 5, 4, 3, 2, 2, 1, 0, 0, 1, 0, 0, 1, 2, 2, 3, 2, 2, 1, 0, 0, 1, 0, 0, 1, 2, 2, 3, 4, 5, 6, 7, 7, 8, 7, 7, 8, 9, 9, 10, 11, 12, 13, 14

Offset: 0

Rémy Sigrist, Feb 07 2020

Looks much like A005536, in particular in respect of its symmetries of scale (compare the scatterplots). - Peter Munn, Jun 21 2021

Rémy Sigrist, Table of n, a(n) for n = 0..16384
Larry Riddle, Koch Curve
Rémy Sigrist, PARI program for A332205
Index entries for sequences related to coordinates of 2D curves

Cf. A005536, A007052, A065359, A332204 (real part and additional comments), A332206 (positions of 0's, cf. A001196).

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

PARI
```
\\ See Links section.
```

a(2^(2*k-1)) = A007052(k) for any k >= 0.

a(4^k-m) = a(m) for any k >= 0 and m = 0..4^k.

A335359 a(n) is the Y-coordinate of the n-th point of the Koch curve; sequence A335358 gives X-coordinates.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 2, 2, 3, 2, 2, 1, 0, 0, 1, 0, 0, 1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 6, 6, 7, 8, 8, 9, 8, 8, 7, 6, 6, 7, 6, 6, 5, 4, 4, 3, 2, 2, 1, 0, 0, 1, 0, 0, 1, 2, 2, 3, 2, 2, 1, 0, 0, 1, 0, 0, 1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 6, 6, 7, 8, 8, 9, 10, 10, 11, 12

Offset: 0

Rémy Sigrist, Jun 03 2020

Coordinates are given on a hexagonal lattice with X-axis and Y-axis as follows:
           Y
          /
         /
        0 ---- X

			The Koch curve starts (on a hexagonal lattice) as follows:
   .   .   .   .   .   .   +   .   .   .   .   .   .
                          /8\
     .   .   .   .   +---+   +---+   .   .   .   .
                     6\  7   9  /10
   .   .   .   +   .   +   .   +   .   +   .   .   .
              /2\     /5        \     / \
     .   +---+   +---+   .   .   +---+   +---+   .
         0   1   3   4           12  13  15  16
Hence, a(6) = a(7) = a(9) = a(10) = 2.

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Wikipedia, Koch snowflake
Index entries for sequences related to coordinates of 2D curves

Cf. A065359, A332206, A335358.

{ hex = [1,I,I-1,-1,-I,1-I]; z=0; for (n=0, 84, print1 (imag(z)", "); q=digits(n, 4); d=sum(k=1, #q, if (q[k]==1, +1, q[k]==2, -1, 0)); z+=hex[1+d%#hex]) }

a(n) = 0 iff n belongs to A332206.

cp's OEIS Frontend

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).

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A332205 a(n) is the imaginary 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).

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Author

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Links

Crossrefs

Programs

Mathematica

PARI

Formula

A335359 a(n) is the Y-coordinate of the n-th point of the Koch curve; sequence A335358 gives X-coordinates.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula