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).

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, 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, 13, 10, 10, 12, 11, 11, 12, 11, 11, 11, 14, 11, 13, 12, 13

Offset: 1

Author

Rémy Sigrist, Dec 31 2019

Keywords

• nonn

Comments

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.

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).

Examples

			The first terms, alongside the corresponding values of f(n), are:
  n   a(n)  f(n)
  --  ----  ------
   0  N/A        0
   1     1       i
   2     2    -2+i
   3     2    -2-i
   4     3     1-i
   5     3   1+2*i
   6     3  -2+2*i
   7     4  -2-2*i
   8     4   2-2*i
   9     4   2+2*i
  10     5  -3+2*i
  11     4  -3-2*i
  12     6   3-2*i
See also illustration in Links section.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, Illustration of first steps
• Rémy Sigrist, Colored representation of f(n) for n = 0..1000000 in the complex plane (where the color is function of n)
• Rémy Sigrist, Colored representation of the variant where the value v can appear up to 3*v times
• Rémy Sigrist, Colored representation of the variant where the value v can appear up to A000265(v) times
• Rémy Sigrist, Colored representation of the variant where the value v can appear up to prime(v) times
• Rémy Sigrist, PARI program for A330779

Crossrefs

See A331002 and A331003 for the real and imaginary parts of f, respectively.

See A330780 for another variant.

Cf. A000265.

Programs

PARI

See Links section.