A330780 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).

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, 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, 6, 6, 6, 6, 8, 6, 6, 6, 6, 6, 6, 8, 6, 6, 6, 6, 6, 6, 7

Offset

1,2

Comments

The sequence {m(k)} is 8-periodic:

m(1) = 2 + i,

m(2) = 1 + 2*i, m(3) | m(2)

m(3) = -1 + 2*i, * | *

m(4) = -2 + i, m(4) * | * m(1)

m(5) = -2 - i, ------+------

m(6) = -1 - 2*i, m(5) * | * m(8)

m(7) = 1 - 2*i, * | *

m(8) = 2 - i. m(6) | m(7)

Links

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Rémy Sigrist, Illustration of first steps

Rémy Sigrist, 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 v^3 times

Rémy Sigrist, PARI program for A330780

Example

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

Prog

(PARI) See Links section.

Crossrefs

See A331004 and A331005 for the real and imaginary parts of f, respectively.

See A330779 for another variant.

Sequence in context: A066490 A156685 A124230 * A090973 A076634 A083277

Adjacent sequences: A330777 A330778 A330779 * A330781 A330782 A330783

Keyword

nonn

Author

Rémy Sigrist, Dec 31 2019

Status

approved