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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. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Rotations and reflections of a placement are counted.

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

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).

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)

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 *)

CROSSREFS

Cf. A296468, A296997.

Sequence in context: A166817 A166799 A001561 * A060094 A201062 A240259

Adjacent sequences:  A296995 A296996 A296997 * A296999 A297000 A297001

KEYWORD

nonn,easy

AUTHOR

Heinrich Ludwig, Dec 23 2017

STATUS

approved

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Last modified May 12 13:46 EDT 2021. Contains 343823 sequences. (Running on oeis4.)