A330780 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

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