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.
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
Links
- 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).
Programs
-
Mathematica
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 *)
Formula
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)
Comments