A279437 Number of ways to place 3 points on an n X n square grid so that no more than 2 points are on a vertical or horizontal straight line.

0, 4, 78, 528, 2200, 6900, 17934, 40768, 83808, 159300, 284350, 482064, 782808, 1225588, 1859550, 2745600, 3958144, 5586948, 7739118, 10541200, 14141400, 18711924, 24451438, 31587648, 40380000, 51122500, 64146654, 79824528, 98571928, 120851700, 147177150, 178115584

Offset

1,2

Comments

Column 4 of triangle A279445.

Rotations and reflections of placements are counted. For numbers if they are to be ignored see A279447.

For condition "no more than 2 points on straight lines at any angle", see A045996.

Links

Heinrich Ludwig, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Formula

a(n) = (n^6 - 5*n^4 + 6*n^3 - 2*n^2)/6.

a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7).

G.f.: 2*x^2*(2 + 25*x + 33*x^2 + x^3 - x^4) / (1 - x)^7. - Colin Barker, Dec 12 2016

Mathematica

Table[(n^6 - 5 n^4 + 6 n^3 - 2 n^2)/6, {n, 32}] (* or *)
Rest@ CoefficientList[Series[2 x^2*(2 + 25 x + 33 x^2 + x^3 - x^4)/(1 - x)^7, {x, 0, 32}], x] (* Michael De Vlieger, Dec 12 2016 *)

Prog

(PARI) concat(0, Vec(2*x^2*(2 + 25*x + 33*x^2 + x^3 - x^4) / (1 - x)^7 + O(x^50))) \\ Colin Barker, Dec 12 2016

Crossrefs

Cf. A045996, A249447, A279444, A279445, A197458.

Same problem but 2, 4..9 points: A083374, A279438, A279439, A279440, A279441, A279442, A279443.

Sequence in context: A370953 A201984 A210519 * A296997 A220240 A246606

Adjacent sequences: A279434 A279435 A279436 * A279438 A279439 A279440

Keyword

nonn,easy

Author

Heinrich Ludwig, Dec 12 2016

Status

approved