A332204 - OEIS

%I #43 Aug 30 2024 10:18:56

%S 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,

%T 20,21,22,22,23,24,25,26,26,27,28,29,30,31,31,32,31,31,32,33,33,34,35,

%U 36,37,38,39,39,40,41,42,43,43,44,45,46,47,48,49,49,50

%N 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).

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

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

%C          3 _     2     _ 1

%C             \_   |   _/

%C               \_ | _/

%C                 \|/

%C          4 ------.------ 0

%C                _/|\_

%C              _/  |  \_

%C            _/    |    \_

%C          5       6       7

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

%C - start at the origin looking to the right,

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

%C     - move forward to the next lattice point

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

%C     - if A096268(k)=0

%C       then turn 45 degrees to the left

%C       otherwise turn 90 degrees to the right,

%C - this connects the first differences of A065359 and A096268.

%H Rémy Sigrist, <a href="/A332204/b332204.txt">Table of n, a(n) for n = 0..16384</a>

%H Larry Riddle, <a href="http://ecademy.agnesscott.edu/~lriddle/ifs/kcurve/kcurve.htm">Koch Curve</a>

%H Rémy Sigrist, <a href="/A332204/a332204_1.png">Illustration of first terms</a>

%H Rémy Sigrist, <a href="/A332204/a332204_2.png">Representation of f(n) in the complex plan for n = 0..2^14</a>

%H Rémy Sigrist, <a href="/A332204/a332204.gp.txt">PARI program for A332204</a>

%H <a href="/index/Con#coordinates_2D_curves">Index entries for sequences related to coordinates of 2D curves</a>

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

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

%e The first terms, alongside f(n) and A065359(n), are:

%e   n   a(n)  f(n)   A065359(n)

%e   --  ----  -----  ----------

%e    0     0      0           0

%e    1     1      1           1

%e    2     2    2+i          -1

%e    3     3      3           0

%e    4     4      4           1

%e    5     5    5+i           2

%e    6     5  5+2*i           0

%e    7     6  6+2*i           1

%e    8     7  7+3*i          -1

%e    9     8  8+2*i           0

%e   10     9  9+2*i          -2

%e   11     9    9+i          -1

%e   12    10     10           0

%e   13    11     11           1

%e   14    12   12+i          -1

%e   15    13     13           0

%e   16    14     14           1

%t A065359[0] = 0;

%t A065359[n_] := -Total[(-1)^PositionIndex[Reverse[IntegerDigits[n, 2]]][1]];

%t g[z_] := z/GCD[Re[z], Im[z]];

%t Module[{n = 0}, Re[NestList[# + g[(1+I)^A065359[n++]] &, 0, 100]]] (* _Paolo Xausa_, Aug 28 2024 *)

%o (PARI) \\ See Links section.

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

%K nonn,base

%O 0,3

%A _Rémy Sigrist_, Feb 07 2020