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

Author

Rémy Sigrist, Dec 31 2019

Keywords

• nonn

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)

Examples

			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.
		

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

Crossrefs

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

See A330779 for another variant.

Programs

PARI

See Links section.