A296468 -id:A296468 - cp's OEIS frontend

A296997 Number of ways to place 3 points on an n X n point grid so that no point is equally distant from two other points on the same row or the same column.

0, 4, 78, 544, 2260, 7068, 18298, 41472, 85032, 161300, 287430, 486624, 789308, 1234604, 1871730, 2761728, 3979088, 5613732, 7772862, 10583200, 14193060, 18774844, 24527338, 31678464, 40487800, 51249588, 64295478, 79997792, 98772492, 121082700, 147441890, 178417664

Offset: 1

Heinrich Ludwig, Dec 23 2017

Rotations and reflections of a placement are counted. If they are to be ignored, see A296996.

The condition of placements is also known as "no 3-term arithmetic progressions".

Heinrich Ludwig, Table of n, a(n) for n = 1..256
Index entries for linear recurrences with constant coefficients, signature (5,-8,0,14,-14,0,8,-5,1).

Cf. A296468, A296996, A296998.

Array[(#^6 - 3 #^4 - 3 #^3 + 8 #^2)/6 - # Boole[OddQ@ #]/2 &, 32] (* Michael De Vlieger, Dec 23 2017 *)
CoefficientList[ Series[-2x (2 + 29x + 93x^2 + 82x^3 + 32x^4 + x^5 + x^6)/((x - 1)^7 (x + 1)^2), {x, 0, 31}], x] (* or *)
LinearRecurrence[{5, -8, 0, 14, -14, 0, 8, -5, 1}, {0, 4, 78, 544, 2260, 7068, 18298, 41472, 85032}, 32] (* Robert G. Wilson v, Jan 15 2018 *)

concat(0, Vec(2*x^2*(2 + 29*x + 93*x^2 + 82*x^3 + 32*x^4 + x^5 + x^6) / ((1 - x)^7*(1 + x)^2) + O(x^40))) \\ Colin Barker, Dec 23 2017

a(n) = (n^6 - 3*n^4 - 3*n^3 + 8*n^2)/6 - (n == 1 (mod 2))*n/2.
a(n) = (n^6 - 3*n^4 - 3*n^3 + 8*n^2)/6 for n even,
a(n) = (n^6 - 3*n^4 - 3*n^3 + 8*n^2 - 3*n)/6 for n odd.
From Colin Barker, Dec 23 2017: (Start)
G.f.: 2*x^2*(2 + 29*x + 93*x^2 + 82*x^3 + 32*x^4 + x^5 + x^6) / ((1 - x)^7*(1 + x)^2).
a(n) = 5*a(n-1) - 8*a(n-2) + 14*a(n-4) - 14*a(n-5) + 8*a(n-7) - 5*a(n-8) + a(n-9) for n>9.
(End)

A296998 Number of ways to place 4 points on an n X n point grid so that no point is equally distant from two other points on the same row or the same column.

Original entry on oeis.org

0, 1, 90, 1620, 11810, 56613, 206234, 623904, 1641654, 3882985, 8431280, 17078364, 32641102, 59401153, 103638420, 174341920, 284041304, 449881893, 694849380, 1049316180, 1552766796, 2255936441, 3223157762, 4535226864, 6292505300, 8618661337, 11664674406, 15613614884

Offset: 1

Heinrich Ludwig, Dec 23 2017

Rotations and reflections of a placement are counted.

The condition of placements is also known as "no 3-term arithmetic progressions".

Heinrich Ludwig, Table of n, a(n) for n = 1..256
Index entries for linear recurrences with constant coefficients, signature (3,-1,-3,1,-3,8,0,-2,-2,-10,10,2,2,0,-8,3,-1,3,1,-3,1).

Cf. A296468, A296997.

Array[Binomial[#^2, 4] - 2 # (Floor[(# - 1)^2/4] (#^2 - 3) - (5 #^2/12 - 3 #/2 + 1/3 - Boole[Divisible[#, 3]]/3 + 3 Boole[OddQ@ #]/4 + Boole[Mod[#, 4] == 2])) &, 28] (* Michael De Vlieger, Dec 23 2017 *)

a(n) = binomial(n^2, 4) - (floor((n-1)^2/4)*(n^2-3) - ((5/12)*n^2 - (3/2)*n + 1/3 + (n == 0 mod 3)*(-1/3) + (n == 1 mod 2)*3/4 + (n == 2 mod 4)))*2*n.
a(n) = (n^8 -6*n^6 -12*n^5 +35*n^4 +56*n^3 -150*n^2)/24 + b(n), where
  b(n) = 0              for n == 0           mod 12,
  b(n) = -n^3/2 +11*n/3 for n == 1, 5, 7, 11 mod 12,
  b(n) = 8*n/3          for n == 2, 10       mod 12,
  b(n) = -n^3/2 +3*n    for n == 3, 9        mod 12,
  b(n) = 2*n/3          for n == 4, 8        mod 12,
  b(n) = 2*n            for n == 6           mod 12.
Conjectures from Colin Barker, Dec 23 2017: (Start)
G.f.: x^2*(1 + 87*x + 1351*x^2 + 7043*x^3 + 23072*x^4 + 52978*x^5 + 95887*x^6 + 138345*x^7 + 166488*x^8 + 164998*x^9 + 137795*x^10 + 94181*x^11 + 52940*x^12 + 23010*x^13 + 7601*x^14 + 1647*x^15 + 251*x^16 + 15*x^17 - 10*x^18) / ((1 - x)^9*(1 + x)^4*(1 + x^2)^2*(1 + x + x^2)^2).
a(n) = 3*a(n-1) - a(n-2) - 3*a(n-3) + a(n-4) - 3*a(n-5) + 8*a(n-6) - 2*a(n-8) - 2*a(n-9) - 10*a(n-10) + 10*a(n-11) + 2*a(n-12) + 2*a(n-13) - 8*a(n-15) + 3*a(n-16) - a(n-17) + 3*a(n-18) + a(n-19) - 3*a(n-20) + a(n-21) for n>21.
(End)

A300131 Largest number of points that can be placed on an n X n point grid so that no point is equally distant from two other points on the same straight line.

Original entry on oeis.org

1, 4, 6, 9, 16, 17, 21, 26, 31, 34, 40, 46

Offset: 1

Heinrich Ludwig, Feb 26 2018

This definition is a 2-dimensional generalization of A003002 ("no 3-term arithmetic progressions"). It generalizes the definition of A296468 to include not only triples on horizontal or vertical lines but on any straight line.

			On a 10 X 10 point grid 34 points (X) can be placed at most. Example:
  . X . . . X X . . .
  X X . . . X X . X .
  X . X X . . . . X .
  . . X . . . . X . X
  . . . . . . . . X X
  X X . . . . . . . .
  X . X . . . . X . .
  . X . . . . X X . X
  . X . X X . . . X X
  . . . X X . . . X .

Cf. A003002, A296468.

a(11) from Bert Dobbelaere, Jan 09 2020

a(12) from Bert Dobbelaere, Jan 12 2020

A296994 Largest number of points that can be selected from an n X n X n triangular point grid so that no selected point is equally distant from two other selected points on a straight line, which is parallel to one side of the grid.

Original entry on oeis.org

1, 3, 4, 7, 10, 14, 18, 20, 23, 27, 31, 36, 42, 48, 54, 61, 68, 76, 84, 92, 98

Offset: 1

Heinrich Ludwig, Mar 26 2018

This sequence generalizes the idea of A003002 ("no 3-term arithmetic progressions") for triangular point grids.

For the same idea applied to square grids see A296468 and A300131.

			At most 54 points (X) can be chosen from a 15 X 15 X 15 triangular point grid under the condition mentioned above. Example:
                 o
                X X
               X o X
              o X X o
             X X o X X
            X o o o o X
           o o X o X o o
          o o o o o o o o
         o o X X o X X o o
        o X o o X X o o X o
       X X o o X o X o o X X
      X o X X o o o o X X o X
     o X X o o X o X o o X X o
    X X o X X o o o o X X o X X
   X o o o o X X o X X o o o o X

Cf. A296468, A300131.

a(20) from Heinrich Ludwig, Apr 24 2018

a(21) from Heinrich Ludwig, May 01 2018

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A296997 Number of ways to place 3 points on an n X n point grid so that no point is equally distant from two other points on the same row or the same column.

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A296998 Number of ways to place 4 points on an n X n point grid so that no point is equally distant from two other points on the same row or the same column.

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A300131 Largest number of points that can be placed on an n X n point grid so that no point is equally distant from two other points on the same straight line.

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A296994 Largest number of points that can be selected from an n X n X n triangular point grid so that no selected point is equally distant from two other selected points on a straight line, which is parallel to one side of the grid.

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