A296997 -id:A296997 - cp's OEIS frontend

A296996 Number of nonequivalent (mod D_8) 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, 1, 14, 75, 310, 911, 2373, 5254, 10824, 20305, 36300, 61081, 99294, 154735, 234955, 345836, 498848, 702609, 973674, 1324135, 1776950, 2348511, 3069649, 3961970, 5065800, 6408961, 8043048, 10003189, 12354174, 15139615, 18439575, 22307416, 26840704, 32103905, 38214470

Offset: 1

Heinrich Ludwig, Jan 12 2018

Rotations and reflections of placements are not counted. If they are to be counted see A296997.

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,-11,6,14,-14,-6,11,-1,-3,1).

Cf. A296997.

Array[(#^6 - 3 #^4 + 5 #^3 - 4 #^2 + 4 #)/48 + Boole[OddQ@ #] (8 #^3 - 18 #^2 + 7 #)/48 &, 35] (* or *)
Rest@ CoefficientList[Series[x^2*(1 + 11 x + 32 x^2 + 82 x^3 + 54 x^4 + 57 x^5 + 2 x^6 + 2 x^7 - x^8)/((1 - x)^7*(1 + x)^4), {x, 0, 35}], x] (* Michael De Vlieger, Jan 12 2018 *)

concat(0, Vec(x^2*(1 + 11*x + 32*x^2 + 82*x^3 + 54*x^4 + 57*x^5 + 2*x^6 + 2*x^7 - x^8) / ((1 - x)^7*(1 + x)^4) + O(x^40))) \\ Colin Barker, Jan 12 2018

a(n) = (n^6 -3*n^4 +5*n^3 -4*n^2 +4n)/48 + (n == 1 mod 2)*(8*n^3 -18n^2 +7*n)/48.
From Colin Barker, Jan 12 2018: (Start)
G.f.: x^2*(1 + 11*x + 32*x^2 + 82*x^3 + 54*x^4 + 57*x^5 + 2*x^6 + 2*x^7 - x^8) / ((1 - x)^7*(1 + x)^4).
a(n) = (n^6 - 3*n^4 + 5*n^3 - 4*n^2 + 4*n) / 48 for n even.
a(n) = (n^6 - 3*n^4 + 13*n^3 - 22*n^2 + 11*n) / 48 for n odd.
a(n) = 3*a(n-1) + a(n-2) - 11*a(n-3) + 6*a(n-4) + 14*a(n-5) - 14*a(n-6) - 6*a(n-7) + 11*a(n-8) - a(n-9) - 3*a(n-10) + a(n-11) for n>11.
(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)

cp's OEIS Frontend

A296996 Number of nonequivalent (mod D_8) 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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula