This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A290536 #21 Dec 22 2020 04:10:43
%S A290536 0,1,2,1,3,2,2,0,4,3,3,0,3,1,1,-2,5,4,4,0,4,1,1,-4,4,2,2,-2,-1,-1,-4,
%T A290536 6,5,5,0,5,1,1,-6,5,2,2,-4,-3,-3,-8,5,3,3,-2,-1,-1,-6,0,-4,-3,-3,-4,7,
%U A290536 6,6,0,6,1,1,-8,6,2,2,-6,-5,-5,-12,6,3,3,-4,-3
%N A290536 Let S be the sequence generated by these rules: 0 is in S, and if z is in S, then z + 1 and z * (1+i) are in S (where i denotes the imaginary unit), and duplicates are deleted as they occur; a(n) = the real part of the n-th term of S.
%C A290536 See A290537 for the imaginary part of the n-th term of S.
%C A290536 See A290538 for the square of the norm of the n-th term of S.
%C A290536 The representation of the first terms of S in the complex plane has nice fractal features (see also Links section).
%C A290536 The sequence S is a "complex" variant of A232559.
%C A290536 The sequence S is a permutation of the Gaussian integers (Z[i]):
%C A290536 - let u be the function defined over Z[i] by z -> z+1,
%C A290536 - let v be the function defined over Z[i] by z -> z*(1+i),
%C A290536 - for m, n, o, p and q >= 0,
%C A290536 let f(m,n,o,p,q) = u^m(v(u^n(v(u^o(v(v(u^p(v(v(u^q(0)))))))))))
%C A290536 (where w^k denotes the k-th iterate of w),
%C A290536 - f(m,0,0,0,0) = m, and any nonnegative integer x can be represented in this way for some m >= 0,
%C A290536 - f(m,n,0,0,0) = m+n + n*i, and any Gaussian integer x+y*i with 0 <= x and 0 <= y <= x can be represented in this way for some m and n >= 0,
%C A290536 - f(m,n,o,0,0) = f(m,n,0,0,0) + 2*o*i, and any Gaussian integer x+y*i with 0 < x and 0 <= y can be represented in this way for some m, n and o >= 0,
%C A290536 - f(m,n,o,p,0) = f(m,n,o,0,0) - 4*p, and any Gaussian integer x+y*i with 0 <= y can be represented in this way for some m, n, o and p >= 0,
%C A290536 - f(m,n,o,p,q) = f(m,n,o,p,0) - 8*q*i, and any Gaussian integer x+y*i can be represented in this way for some m, n, o, p and q >= 0,
%C A290536 - in other words, any Gaussian integer can be reached from 0 after a finite number of steps chosen in { u, v }, QED.
%H A290536 Rémy Sigrist, <a href="/A290536/b290536.txt">Table of n, a(n) for n = 1..10000</a>
%H A290536 Rémy Sigrist, <a href="/A290536/a290536_1.png">Representation of the first 100000 terms of S in the complex plane</a>
%H A290536 Rémy Sigrist, <a href="/A290536/a290536.png">Colorized representation of the first 100000 terms of S in the complex plane</a>
%H A290536 Rémy Sigrist, <a href="/A290536/a290536.gp.txt">PARI program for A290536</a>
%H A290536 Wikipedia, <a href="https://en.wikipedia.org/wiki/Gaussian_integer">Gaussian integer</a>
%e A290536 S(1) = 0 by definition; so a(1) = 0.
%e A290536 S(1)+1 = 1 has not yet occurred; so S(2) = 1 and a(2) = 1.
%e A290536 S(1)*(i+i) = 0 has already occurred.
%e A290536 S(2)+1 = 2 has not yet occurred; so S(3) = 2 and a(3) = 2.
%e A290536 S(2)*(1+i) = 1+i has not yet occurred; so S(4) = 1+i and a(4) = 1.
%e A290536 S(3)+1 = 3 has not yet occurred; so S(5) = 3 and a(5) = 3.
%o A290536 (PARI) See Links section.
%Y A290536 Cf. A232559, A290537, A290538.
%K A290536 sign,look
%O A290536 1,3
%A A290536 _Rémy Sigrist_, Aug 05 2017