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URL: https://oeis.org/A296997

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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.
3
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
1,2
COMMENTS
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".
LINKS
FORMULA
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)
MATHEMATICA
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 *)
PROG
(PARI) 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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Heinrich Ludwig, Dec 23 2017
STATUS
approved