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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. 9
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 (list; graph; refs; listen; history; text; internal format)
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 *)

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

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Last modified December 12 22:06 EST 2019. Contains 329963 sequences. (Running on oeis4.)