%I
%S 1,2,2,2,2,3,3,3,3,3,3,3,4,3,3,4,4,4,4,4,4,4,5,4,4,4,4,4,4,5,4,4,5,5,
%T 5,5,5,5,5,6,5,5,5,5,6,5,5,5,5,5,5,6,5,5,5,5,5,6,5,6,6,6,6,6,6,6,7,6,
%U 6,6,6,6,8,6,6,6,6,6,6,8,6,6,6,6,6,6,7
%N Lexicographically earliest sequence of positive integers such that for any v > 0, the value v appears up to v^2 times, and the associate function f defined by f(n) = Sum_{k = 1..n} a(k) * m(k) for n >= 0 is injective (where {m(k)} corresponds to knight's moves, see Comments for precise definition).
%C The sequence {m(k)} is 8periodic:
%C m(1) = 2 + i,
%C m(2) = 1 + 2*i, m(3)  m(2)
%C m(3) = 1 + 2*i, *  *
%C m(4) = 2 + i, m(4) *  * m(1)
%C m(5) = 2  i, +
%C m(6) = 1  2*i, m(5) *  * m(8)
%C m(7) = 1  2*i, *  *
%C m(8) = 2  i. m(6)  m(7)
%H Rémy Sigrist, <a href="/A330780/b330780.txt">Table of n, a(n) for n = 1..10000</a>
%H Rémy Sigrist, <a href="/A330780/a330780_1.png">Illustration of first steps</a>
%H Rémy Sigrist, <a href="/A330780/a330780.png">Representation of f(n) for n = 0..1000000 in the complex plane</a> (where the color is function of n)
%H Rémy Sigrist, <a href="/A330780/a330780_2.png">Colored representation of the variant where the value v can appear up to v^3 times</a>
%H Rémy Sigrist, <a href="/A330780/a330780.gp.txt">PARI program for A330780</a>
%e The first terms, alongside the correspond value of f(n), are:
%e n a(n) f(n)
%e   
%e 0 N/A 0
%e 1 1 2+i
%e 2 2 4+5*i
%e 3 2 2+9*i
%e 4 2 2+11*i
%e 5 2 6+9*i
%e 6 3 9+3*i
%e 7 3 63*i
%e 8 3 6*i
%e 9 3 63*i
%e 10 3 9+3*i
%e 11 3 6+9*i
%e 12 3 12*i
%e See also illustration in Links section.
%o (PARI) See Links section.
%Y See A331004 and A331005 for the real and imaginary parts of f, respectively.
%Y See A330779 for another variant.
%K nonn
%O 1,2
%A _Rémy Sigrist_, Dec 31 2019
