A279439 Number of ways to place 5 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, 45, 2304, 34020, 270720, 1475145, 6209280, 21654864, 65422080, 176467005, 434206080, 990140580, 2117816064, 4288771305, 8284308480, 15355471680, 27446584320, 47501098029, 79872376320, 130866406020, 209448328320, 328150139625, 504222960384, 761083938000

Offset

1,3

Comments

Column 6 of triangle A279445.

Rotations and reflections of placements are counted. For numbers if they are to be ignored see A279449.

For condition "no more than 2 points on straight lines at any angle", see A194190.

Links

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

Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Formula

a(n) = (n^10 -30*n^8 +90*n^7 -27*n^6 -270*n^5 +500*n^4 -360*n^3 +96*n^2)/120.

a(n) = 11*a(n-1) -55*a(n-2) +165*a(n-3) -330*a(n-4) +462*a(n-5) -462*a(n-6) +330*a(n-7) -165*a(n-8) +55*a(n-9) -11*a(n-10) +*a(n-11).

G.f.: 9*x^3*(5 +201*x +1239*x^2 +1755*x^3 +335*x^4 -165*x^5 -11*x^6 +x^7) / (1 -x)^11. - Colin Barker, Dec 22 2016

Mathematica

Table[(n^10 - 30 n^8 + 90 n^7 - 27 n^6 - 270 n^5 + 500 n^4 - 360 n^3 + 96 n^2)/120, {n, 25}] (* or *)
Rest@ CoefficientList[Series[9 x^3*(5 + 201 x + 1239 x^2 + 1755 x^3 + 335 x^4 - 165 x^5 - 11 x^6 + x^7)/(1 - x)^11, {x, 0, 25}], x] (* Michael De Vlieger, Dec 22 2016 *)

Prog

(PARI) concat(vector(2), Vec(9*x^3*(5 +201*x +1239*x^2 +1755*x^3 +335*x^4 -165*x^5 -11*x^6 +x^7) / (1 -x)^11 + O(x^30))) \\ Colin Barker, Dec 22 2016

Crossrefs

Cf. A194190, A279449, A279444, A279445, A197458.

Same problem but 2,3,4,6..9 points: A083374, A279437, A279438, A279440, A279441, A279442, A279443.

Sequence in context: A009989 A095658 A342405 * A251443 A109941 A035097

Adjacent sequences: A279436 A279437 A279438 * A279440 A279441 A279442

Keyword

nonn,easy

Author

Heinrich Ludwig, Dec 21 2016

Status

approved