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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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%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