A194190 -id:A194190 - cp's OEIS frontend

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

Heinrich Ludwig, Dec 21 2016

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.

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).

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

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

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 *)

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

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

A194193 Square array read by antidiagonals downwards: T(n,k) = number of ways to arrange k indistinguishable points on an n X n square grid so that no three points are collinear at any angle.

Original entry on oeis.org

1, 0, 4, 0, 6, 9, 0, 4, 36, 16, 0, 1, 76, 120, 25, 0, 0, 78, 516, 300, 36, 0, 0, 28, 1278, 2148, 630, 49, 0, 0, 2, 1668, 9498, 6768, 1176, 64, 0, 0, 0, 998, 25052, 47331, 17600, 2016, 81, 0, 0, 0, 204, 36698, 215448, 175952, 40120, 3240, 100, 0, 0, 0, 11, 26700, 620210

Offset: 1

R. H. Hardin, Aug 18 2011

Columns 4..7 are A175383, A194190, A194191, A194192 respectively. - Heinrich Ludwig, Nov 16 2016

			Table starts:
...1.....0.......0........0..........0...........0............0............0
...4.....6.......4........1..........0...........0............0............0
...9....36......76.......78.........28...........2............0............0
..16...120.....516.....1278.......1668.........998..........204...........11
..25...300....2148.....9498......25052.......36698........26700.........8242
..36...630....6768....47331.....215448......620210......1073076......1035097
..49..1176...17600...175952....1189868.....5367308.....15657764.....28228158
..64..2016...40120...545764....5199888....34678364....159413700....491910848
..81..3240...82608..1461672...18520572...169259212...1108580092...5122725512
.100..4950..157252..3507553...56978440...682686652...6030207624..38914424892
.121..7260..280988..7701638..155627304..2356999994..26852315940.229093733030
.144.10296..477012.15773526..388897892..7294368210.104865006648
.169.14196..775172.30375194..894254904.20227526910
.196.19110.1214768.55695587.1932504496
.225.25200.1844512.97777392
.256.32640.2725000
...
Some solutions for n=4, k=4:
..0..0..1..0....0..0..0..0....0..0..0..0....0..0..1..0....1..0..0..0
..1..0..0..0....1..0..0..0....0..0..1..0....1..0..0..0....0..0..0..1
..0..0..0..0....0..1..0..1....1..0..1..0....1..0..0..0....0..0..0..1
..0..0..1..1....0..1..0..0....0..1..0..0....0..0..0..1....1..0..0..0

R. H. Hardin and Heinrich Ludwig, Table of n, a(n) for n = 1..199, (first 181 terms from R. H. Hardin)

Column 1 is A000290.
Column 2 is A083374.
Column 3 is A045996.
Column 4 is A175383.
Column 5 is A194190.
Column 6 is A194191.
Column 7 is A194192.

A235456 Number of non-equivalent (mod D_4) ways to arrange 5 points on an n X n square grid so that no three points are collinear.

Original entry on oeis.org

5, 217, 3192, 27041, 149217, 650566, 2317137, 7124316, 19459757, 48617666, 111797647, 241575473

Offset: 3

Heinrich Ludwig, Jan 12 2014

Column 5 of A235453.

Without the restriction "non-equivalent (mod D_4)" the numbers are given by A194190, n >= 3.

			There are a(3) = 5 non-equivalent ways to place 5 points (X) on a 3 X 3 grid:
  X . X    X . X    . X X    X X .    X . X
  X . X    X . .    X . X    X . .    X X .
  . X .    . X X    . X .    . X X    . X .

Cf. A235453, A194190, A235454 (3 points), A235455 (4 points), A235457 (6 points), A235458 (7 points)

a(13), a(14) from Heinrich Ludwig, Nov 16 2016

cp's OEIS Frontend

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.

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A194193 Square array read by antidiagonals downwards: T(n,k) = number of ways to arrange k indistinguishable points on an n X n square grid so that no three points are collinear at any angle.

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A235456 Number of non-equivalent (mod D_4) ways to arrange 5 points on an n X n square grid so that no three points are collinear.

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