A194192 -id:A194192

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A279441

Number of ways to place 7 points on an n X n square grid so that no more than 2 points are on a vertical or horizontal straight line.

+10
9

0, 0, 0, 816, 93000, 2602800, 35526120, 309328320, 1972234656, 9989784000, 42369069600, 155993500080, 511660972680, 1524225598896, 4185197289000, 10715254368000, 25817751281280, 58981960615680, 128554066935936, 268691201838000, 540886175310600, 1052558059827120

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

COMMENTS

Column 8 of triangle A279445.

Rotations and reflections of placements are counted. For numbers if they are to be ignored see A279451.

For condition "no more than 2 points on straight lines at any angle", see A194192.

LINKS

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

Index entries for linear recurrences with constant coefficients, signature (15,-105,455,-1365,3003,-5005,6435,-6435,5005,-3003,1365,-455,105,-15,1).

FORMULA

a(n) = (n^14 -91*n^12 +420*n^11 +693*n^10 -10500*n^9 +33647*n^8 -45780*n^7 +5866*n^6 +65940*n^5 -89796*n^4 +50400*n^3 -10800*n^2)/5040.

a(n) = 15*a(n-1) -105*a(n-2) +455*a(n-3) -1365*a(n-4) +3003*a(n-5) -5005*a(n-6) +6435*a(n-7) -6435*a(n-8) +5005*a(n-9) -3003*a(n-10) +1365*a(n-11) -455*a(n-12) +105*a(n-13) -15*a(n-14) +a(n-15).

G.f.: 24*x^4*(34 +3365*x +53895*x^2 +244910*x^3 +355390*x^4 +115542*x^5 -42490*x^6 -11570*x^7 +1500*x^8 +145*x^9 -x^10) / (1 -x)^15. - Colin Barker, Dec 22 2016

MATHEMATICA

Table[(n^14 - 91 n^12 + 420 n^11 + 693 n^10 - 10500 n^9 + 33647 n^8 - 45780 n^7 + 5866 n^6 + 65940 n^5 - 89796 n^4 + 50400 n^3 - 10800 n^2)/5040, {n, 23}] (* or *)

Rest@ CoefficientList[Series[24 x^4*(34 + 3365 x + 53895 x^2 + 244910 x^3 + 355390 x^4 + 115542 x^5 - 42490 x^6 - 11570 x^7 + 1500 x^8 + 145 x^9 - x^10)/(1 - x)^15, {x, 0, 23}], x] (* Michael De Vlieger, Dec 22 2016 *)

PROG

(PARI) concat(vector(3), Vec(24*x^4*(34 +3365*x +53895*x^2 +244910*x^3 +355390*x^4 +115542*x^5 -42490*x^6 -11570*x^7 +1500*x^8 +145*x^9 -x^10) / (1 -x)^15 + O(x^30))) \\ Colin Barker, Dec 22 2016

(PARI) a(n) = (n^12 -91*n^10 +420*n^9 +693*n^8 -10500*n^7 +33647*n^6 -45780*n^5 +5866*n^4 +65940*n^3 -89796*n^2 +50400*n -10800)*n^2/5040 \\ Charles R Greathouse IV, Dec 22 2016

CROSSREFS

Cf. A194192, A279451, A279444, A279445, A197458.

Same problem but 2..6,8,9 points: A083374, A279437, A279438, A279439, A279440, A279442, A279443.

KEYWORD

nonn,easy

AUTHOR

Heinrich Ludwig, Dec 22 2016

STATUS

approved

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.

+10
7

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

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

LINKS

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

EXAMPLE

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

CROSSREFS

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.

KEYWORD

nonn,tabl

AUTHOR

R. H. Hardin, Aug 18 2011

STATUS

approved

A235458

Number of non-equivalent (mod D_4) ways to arrange 7 points on an n X n square grid so that no three points are collinear.

+10
6

28, 3385, 134353, 1958674, 19929645, 138586349, 753795278, 3356614240, 13108210508, 44374441652, 137349454120

(list; graph; refs; listen; history; text; internal format)

OFFSET

4,1

COMMENTS

Column 7 of A235453.

Without the restriction "non-equivalent (mod D_4)" the numbers are given by A194192, n >= 4.

LINKS

Table of n, a(n) for n=4..14.

EXAMPLE

There are a(4) = 28 non-equivalent ways to place 7 points (X) on a 4 X 4 grid. Example:

. X X .

. . . X

X . . X

X X . .

CROSSREFS

Cf. A235453, A194192, A235454 (3 points), A235455 (4 points), A235456 (5 points), A235457 (6 points).

KEYWORD

nonn,more

AUTHOR

Heinrich Ludwig, Jan 12 2014

EXTENSIONS

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

STATUS

approved

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