A330779 -id:A330779 - cp's OEIS frontend

A331002 a(n) is the real part of f(n) (as defined in A330779).

0, 0, -2, -2, 1, 1, -2, -2, 2, 2, -3, -3, 3, 3, -2, -2, 3, 3, -4, -4, 2, 2, -4, -4, 4, 4, -3, -3, 5, 5, -2, -2, 6, 6, -4, -4, 4, 4, -5, -5, 5, 5, -4, -4, 5, 5, -6, -6, 3, 3, -6, -6, 6, 6, -5, -5, 6, 6, -4, -4, 6, 6, -7, -7, 3, 3, -8, -8, 4, 4, -7, -7, 7, 7, -6

Offset: 0

Rémy Sigrist, Jan 05 2020

sign

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, PARI program for A331002

Cf. A330779.

PARI
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See Links section.
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a(2*k+1) = a(2*k) for any k >= 0.

a(2*k) = a(2*k-1) + A330779(2*k) * (-1)^k for any k > 0.

A331003 a(n) is the imaginary part of f(n) (as defined in A330779).

Original entry on oeis.org

0, 1, 1, -1, -1, 2, 2, -2, -2, 2, 2, -2, -2, 3, 3, -3, -3, 2, 2, -4, -4, 3, 3, -3, -3, 4, 4, -3, -3, 4, 4, -4, -4, 4, 4, -5, -5, 3, 3, -5, -5, 5, 5, -6, -6, 3, 3, -6, -6, 4, 4, -5, -5, 5, 5, -6, -6, 6, 6, -7, -7, 3, 3, -7, -7, 5, 5, -6, -6, 5, 5, -6, -6, 5, 5

Offset: 0

Rémy Sigrist, Jan 05 2020

sign

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, PARI program for A331003

Cf. A330779.

PARI
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See Links section.
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a(2*k+2) = a(2*k+1) for any k >= 0.

a(2*k+1) = a(2*k) + A330779(2*k+1) * (-1)^k for any k >= 0.

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).

Original entry on oeis.org

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

Rémy Sigrist, Dec 31 2019

nonn

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)

			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.

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

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

See A330779 for another variant.

PARI
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See Links section.
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cp's OEIS Frontend

A331002 a(n) is the real part of f(n) (as defined in A330779).

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A331003 a(n) is the imaginary part of f(n) (as defined in A330779).

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