A279438 Number of ways to place 4 points on an n X n square grid so that no more than 2 points are on a vertical or horizontal straight line.

0, 1, 90, 1428, 10600, 51525, 190806, 584080, 1552608, 3701025, 8088850, 16470036, 31616520, 57743413, 101055150, 170433600, 278290816, 441610785, 683206218, 1033218100, 1530887400, 2226630021, 3184447750, 4484709648, 6227340000, 8535450625, 11559457026, 15481719540

Offset

1,3

Comments

Column 5 of triangle A279445.

Rotations and reflections of placements are counted. For numbers if they are to be ignored see A279448.

For condition "no more than 2 points on straight lines at any angle", see A175383.

Links

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

Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Formula

a(n) = (n^8 - 14*n^6 + 30*n^5 - 17*n^4 - 6*n^3 + 6*n^2)/24.

a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9).

G.f.: x^2*(1 + 81*x + 654*x^2 + 904*x^3 + 99*x^4 - 57*x^5 - 2*x^6) / (1 - x)^9. - Colin Barker, Dec 13 2016

Mathematica

Table[(n^8 - 14 n^6 + 30 n^5 - 17 n^4 - 6 n^3 + 6 n^2)/24, {n, 28}] (* Michael De Vlieger, Dec 12 2016 *)
LinearRecurrence[{9, -36, 84, -126, 126, -84, 36, -9, 1}, {0, 1, 90, 1428, 10600, 51525, 190806, 584080, 1552608}, 30] (* Harvey P. Dale, Sep 05 2024 *)

Prog

(PARI) concat(0, Vec(x^2*(1 + 81*x + 654*x^2 + 904*x^3 + 99*x^4 - 57*x^5 - 2*x^6) / (1 - x)^9 + O(x^30))) \\ Colin Barker, Dec 13 2016
(PARI) a(n) = (n^6 - 14*n^4 + 30*n^3 - 17*n^2 - 6*n + 6)*n^2/24 \\ Charles R Greathouse IV, Dec 13 2016

Crossrefs

Cf. A175383, A249448, A279444, A279445, A197458.

Same problem but 2,3,5..9 points: A083374, A279437, A279439, A279440, A279441, A279442, A279443.

Sequence in context: A155016 A179800 A133350 * A250869 A234983 A166817

Adjacent sequences: A279435 A279436 A279437 * A279439 A279440 A279441

Keyword

nonn,easy

Author

Heinrich Ludwig, Dec 12 2016

Status

approved