A296996 -id:A296996 - 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)

A296999 Number of nonequivalent (mod D_8) 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, 17, 226, 1550, 7221, 26120, 78484, 206242, 486640, 1056377, 2137506, 4085167, 7430276, 12964014, 21801632, 35520743, 56249658, 86880957, 131186720, 194133425, 282024809, 402949496, 566950056, 786640454, 1077397347, 1458190435, 1951789266, 2585856152, 3393157995

Offset: 1

Heinrich Ludwig, Jan 21 2018

Rotations and reflections of placements are not counted. If they are to be counted see A296998.

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

Heinrich Ludwig, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-1,-4,4,-4,5,1,-5,6,-10,8,-8,10,-6,5,-1,-5,4,-4,4,1,-3,1).

Cf. A014409, A296996, A296998.

Array[(#^8 - 6 #^6 - 12 #^5 + 64 #^4 + 8 #^3 - 136 #^2 + Boole[OddQ@ #] (14 #^4 - 96 #^3 + 162 #^2 - 92 # + 93))/192 + Boole[Mod[#, 6] == 2] #/6 + Boole[Mod[#, 4] == 2] #/4 + Boole[Mod[#, 6] == 5] (# + 1)/6 &, 30] (* Michael De Vlieger, Jan 21 2018 *)

a(n) = (n^8 - 6*n^6 - 12*n^5 + 64*n^4 + 8*n^3 - 136*n^2 + (n == 1 (mod 2))*(14*n^4 - 96*n^3 + 162*n^2 - 92*n + 93))/192 + (n == 2 (mod 6))*n/6 + (n == 2 (mod 4))*n/4 + (n == 5 (mod 6))*(n + 1)/6.
a(n) = (n^8 - 6*n^6 - 12*n^5)/192 + b(n) + c(n), where
  b(n) = (64*n^4 + 8*n^3 - 136*n^2)/192  for n even,
  b(n) = (78*n^4 - 88*n^3 + 26*n^2 - 92*n + 93)/192  for n odd,
  c(n) = 0          for n == 0, 1, 3, 4, 7, 9  (mod 12),
  c(n) = n/4        for n == 6, 10             (mod 12),
  c(n) = n/6        for n == 8                 (mod 12),
  c(n) = 5/12*n     for n == 2                 (mod 12),
  c(n) = (n + 1)/6  for n == 5, 11             (mod 12).
Conjectures from Colin Barker, Jan 21 2018: (Start)
G.f.: x^2*(1 + 14*x + 176*x^2 + 893*x^3 + 2861*x^4 + 6847*x^5 + 12704*x^6 + 20412*x^7 + 27052*x^8 + 33142*x^9 + 33910*x^10 + 33289*x^11 + 26586*x^12 + 20709*x^13 + 12212*x^14 + 7178*x^15 + 2639*x^16 + 1094*x^17 + 134*x^18 + 68*x^19 - 3*x^20 + 2*x^21) / ((1 - x)^9*(1 + x)^5*(1 - x + x^2)*(1 + x^2)^2*(1 + x + x^2)^2).
a(n) = 3*a(n-1) - a(n-2) - 4*a(n-3) + 4*a(n-4) - 4*a(n-5) + 5*a(n-6) + a(n-7) - 5*a(n-8) + 6*a(n-9) - 10*a(n-10) + 8*a(n-11) - 8*a(n-13) + 10*a(n-14) - 6*a(n-15) + 5*a(n-16) - a(n-17) - 5*a(n-18) + 4*a(n-19) - 4*a(n-20) + 4*a(n-21) + a(n-22) - 3*a(n-23) + a(n-24) for n>24.
(End)

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.

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