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A330779 Lexicographically earliest sequence of positive integers such that for any v > 0, the value v appears up to v times, and the associate function f defined by f(n) = Sum_{k = 1..n} a(k) * i^k for n >= 0 is injective (where i denotes the imaginary unit).

%I A330779 #19 Jan 06 2020 09:14:53
%S A330779 1,2,2,3,3,3,4,4,4,5,4,6,5,5,6,5,5,7,6,6,7,6,6,8,7,7,7,8,7,7,8,8,8,10,
%T A330779 9,8,8,9,8,10,10,9,11,9,9,11,9,9,10,9,9,12,10,11,11,11,12,10,13,10,10,
%U A330779 13,10,10,12,11,11,12,11,11,11,14,11,13,12,13
%N A330779 Lexicographically earliest sequence of positive integers such that for any v > 0, the value v appears up to v times, and the associate function f defined by f(n) = Sum_{k = 1..n} a(k) * i^k for n >= 0 is injective (where i denotes the imaginary unit).
%C A330779 The variant of this sequence where each value can only appear up to once, twice or three times corresponds to A000027, A008619 and A008620 respectively.
%C A330779 Graphically, the representation of f resembles a windmill; the variant of f where we allow the value v to appear 3*v times resembles a butterfly (see illustrations in Links section).
%H A330779 Rémy Sigrist, <a href="/A330779/b330779.txt">Table of n, a(n) for n = 1..10000</a>
%H A330779 Rémy Sigrist, <a href="/A330779/a330779_4.png">Illustration of first steps</a>
%H A330779 Rémy Sigrist, <a href="/A330779/a330779.png">Colored representation of f(n) for n = 0..1000000 in the complex plane</a> (where the color is function of n)
%H A330779 Rémy Sigrist, <a href="/A330779/a330779_1.png">Colored representation of the variant where the value v can appear up to 3*v times</a>
%H A330779 Rémy Sigrist, <a href="/A330779/a330779_2.png">Colored representation of the variant where the value v can appear up to A000265(v) times</a>
%H A330779 Rémy Sigrist, <a href="/A330779/a330779_3.png">Colored representation of the variant where the value v can appear up to prime(v) times</a>
%H A330779 Rémy Sigrist, <a href="/A330779/a330779.gp.txt">PARI program for A330779</a>
%e A330779 The first terms, alongside the corresponding values of f(n), are:
%e A330779   n   a(n)  f(n)
%e A330779   --  ----  ------
%e A330779    0  N/A        0
%e A330779    1     1       i
%e A330779    2     2    -2+i
%e A330779    3     2    -2-i
%e A330779    4     3     1-i
%e A330779    5     3   1+2*i
%e A330779    6     3  -2+2*i
%e A330779    7     4  -2-2*i
%e A330779    8     4   2-2*i
%e A330779    9     4   2+2*i
%e A330779   10     5  -3+2*i
%e A330779   11     4  -3-2*i
%e A330779   12     6   3-2*i
%e A330779 See also illustration in Links section.
%o A330779 (PARI) See Links section.
%Y A330779 See A331002 and A331003 for the real and imaginary parts of f, respectively.
%Y A330779 See A330780 for another variant.
%Y A330779 Cf. A000265.
%K A330779 nonn
%O A330779 1,2
%A A330779 _Rémy Sigrist_, Dec 31 2019