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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 (list; graph; refs; listen; history; text; internal format)
 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 Heinrich Ludwig, Table of n, a(n) for n = 1..256 Index entries for linear recurrences with constant coefficients, signature (5,-8,0,14,-14,0,8,-5,1). 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 Cf. A296468, A296996, A296998. Sequence in context: A201984 A210519 A279437 * A220240 A246606 A210903 Adjacent sequences:  A296994 A296995 A296996 * A296998 A296999 A297000 KEYWORD nonn,easy AUTHOR Heinrich Ludwig, Dec 23 2017 STATUS approved

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Last modified May 17 06:30 EDT 2021. Contains 343965 sequences. (Running on oeis4.)