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A296999
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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.
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1
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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
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OFFSET
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1,3
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COMMENTS
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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".
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LINKS
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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).
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FORMULA
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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).
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)
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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