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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. 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.

Sequence in context: A250803 A251205 A064583 * A221748 A105989 A266059

Adjacent sequences:  A279438 A279439 A279440 * A279442 A279443 A279444

KEYWORD

nonn,easy

AUTHOR

Heinrich Ludwig, Dec 22 2016

STATUS

approved

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Last modified June 20 01:36 EDT 2019. Contains 324223 sequences. (Running on oeis4.)