A279440 Number of ways to place 6 points on an n X n square grid so that no more than 2 points are on a vertical or horizontal straight line.

0, 0, 6, 2040, 71400, 1005720, 8421630, 50092896, 233483040, 905925600, 3045791430, 9125544120, 24868110696, 62593429080, 147255640350, 326843422080, 689604309120, 1391614736256, 2699616160710, 5055848825400, 9173923662120, 16177675640280, 27798546316926, 46651469520480

Offset

1,3

Comments

Column 7 of triangle A279445.

Rotations and reflections of placements are counted. For numbers if they are to be ignored see A279450.

For condition "no more than 2 points on straight lines at any angle", see A194191.

Links

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

Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Formula

a(n) = (n^12 -55*n^10 +210*n^9 +93*n^8 -2220*n^7 +5855*n^6 -7350*n^5 +4786*n^4 -1440*n^3 +120*n^2)/720.

a(n) = 13*a(n-1) -78*a(n-2) +286*a(n-3) -715*a(n-4) +1287*a(n-5) -1716*a(n-6) +1716*a(n-7) -1287*a(n-8) +715*a(n-9) -286*a(n-10) +78*a(n-11) -13*a(n-12) +*a(n-13).

G.f.: 6*x^3*(1 +327*x +7558*x^2 +39154*x^3 +56220*x^4 +14724*x^5 -6262*x^6 -978*x^7 +131*x^8 +5*x^9) / (1 -x)^13. - Colin Barker, Dec 22 2016

Mathematica

Table[(n^12 - 55 n^10 + 210 n^9 + 93 n^8 - 2220 n^7 + 5855 n^6 - 7350 n^5 + 4786 n^4 - 1440 n^3 + 120 n^2)/720, {n, 24}] (* or *)
Rest@ CoefficientList[Series[6 x^3 (1 + 327 x + 7558 x^2 + 39154 x^3 + 56220 x^4 + 14724 x^5 - 6262 x^6 - 978 x^7 + 131 x^8 + 5 x^9)/(1 - x)^13, {x, 0, 24}], x] (* Michael De Vlieger, Dec 22 2016 *)

Prog

(PARI) concat(vector(2), Vec(6*x^3*(1 +327*x +7558*x^2 +39154*x^3 +56220*x^4 +14724*x^5 -6262*x^6 -978*x^7 +131*x^8 +5*x^9) / (1 -x)^13 + O(x^30))) \\ Colin Barker, Dec 22 2016

Crossrefs

Cf. A194191, A279450, A279444, A279445, A197458.

Same problem but 2..5,7,8,9 points: A083374, A279437, A279438, A279439, A279441, A279442, A279443.

Sequence in context: A209737 A209458 A209520 * A004817 A089535 A059081

Adjacent sequences: A279437 A279438 A279439 * A279441 A279442 A279443

Keyword

nonn,easy

Author

Heinrich Ludwig, Dec 22 2016

Status

approved