A279440 -id:A279440 - cp's OEIS frontend

A279445 Triangle read by rows: T(n, k) is the number of ways to place k points on an n X n square grid so that no more than 2 points are on a vertical or horizontal straight line.

1, 1, 1, 4, 6, 4, 1, 1, 9, 36, 78, 90, 45, 6, 1, 16, 120, 528, 1428, 2304, 2040, 816, 90, 1, 25, 300, 2200, 10600, 34020, 71400, 93000, 67950, 22650, 2040, 1, 36, 630, 6900, 51525, 270720, 1005720, 2602800, 4531950, 4987800, 3110940, 888840, 67950, 1, 49, 1176, 17934

Offset: 1

Heinrich Ludwig, Dec 17 2016

Length of n-th row is A272651(n) + 1, where A272651(n) is the maximal number of points to be placed under the condition mentioned.
Rotations and reflections of placements are counted. If they are to be ignored, see A279453.
For condition "no more than 2 points on a straight line at any angle", see A194193 (but that one is read by antidiagonals).

			The table begins with T(1, 0):
1  1
1  4   6    4     1
1  9  36   78    90    45     6
1 16 120  528  1428  2304  2040   816    90
1 25 300 2200 10600 34020 71400 93000 67950 22650 2040
...
T(3, 2) = 36 because there are 36 ways to place 2 points on a 3 X 3 square grid so that no more than 2 points are on a vertical or horizontal straight line.

Heinrich Ludwig, Table of n, a(n) for n = 1..109

Row sums give A197458.
Columns 2..10: A000290, A083374, A279437, A279438, A279439, A279440, A279441, A279442, A279443.
Diagonal T(n, n) is A279444.
Cf. A279453, A194193.

A279437 Number of ways to place 3 points on an n X n square grid so that no more than 2 points are on a vertical or horizontal straight line.

Original entry on oeis.org

0, 4, 78, 528, 2200, 6900, 17934, 40768, 83808, 159300, 284350, 482064, 782808, 1225588, 1859550, 2745600, 3958144, 5586948, 7739118, 10541200, 14141400, 18711924, 24451438, 31587648, 40380000, 51122500, 64146654, 79824528, 98571928, 120851700, 147177150, 178115584

Offset: 1

Heinrich Ludwig, Dec 12 2016

Column 4 of triangle A279445.
Rotations and reflections of placements are counted. For numbers if they are to be ignored see A279447.
For condition "no more than 2 points on straight lines at any angle", see A045996.

Heinrich Ludwig, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A045996, A249447, A279444, A279445, A197458.

Same problem but 2, 4..9 points: A083374, A279438, A279439, A279440, A279441, A279442, A279443.

Table[(n^6 - 5 n^4 + 6 n^3 - 2 n^2)/6, {n, 32}] (* or *)
Rest@ CoefficientList[Series[2 x^2*(2 + 25 x + 33 x^2 + x^3 - x^4)/(1 - x)^7, {x, 0, 32}], x] (* Michael De Vlieger, Dec 12 2016 *)

concat(0, Vec(2*x^2*(2 + 25*x + 33*x^2 + x^3 - x^4) / (1 - x)^7 + O(x^50))) \\ Colin Barker, Dec 12 2016

a(n) = (n^6 - 5*n^4 + 6*n^3 - 2*n^2)/6.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7).
G.f.: 2*x^2*(2 + 25*x + 33*x^2 + x^3 - x^4) / (1 - x)^7. - Colin Barker, Dec 12 2016

A279438 Number of ways to place 4 points on an n X n square grid so that no more than 2 points are on a vertical or horizontal straight line.

Original entry on oeis.org

0, 1, 90, 1428, 10600, 51525, 190806, 584080, 1552608, 3701025, 8088850, 16470036, 31616520, 57743413, 101055150, 170433600, 278290816, 441610785, 683206218, 1033218100, 1530887400, 2226630021, 3184447750, 4484709648, 6227340000, 8535450625, 11559457026, 15481719540

Offset: 1

Heinrich Ludwig, Dec 12 2016

Column 5 of triangle A279445.
Rotations and reflections of placements are counted. For numbers if they are to be ignored see A279448.
For condition "no more than 2 points on straight lines at any angle", see A175383.

Heinrich Ludwig, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A175383, A249448, A279444, A279445, A197458.

Same problem but 2,3,5..9 points: A083374, A279437, A279439, A279440, A279441, A279442, A279443.

Table[(n^8 - 14 n^6 + 30 n^5 - 17 n^4 - 6 n^3 + 6 n^2)/24, {n, 28}] (* Michael De Vlieger, Dec 12 2016 *)
LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{0,1,90,1428,10600,51525,190806,584080,1552608},30] (* Harvey P. Dale, Sep 05 2024 *)

concat(0, Vec(x^2*(1 + 81*x + 654*x^2 + 904*x^3 + 99*x^4 - 57*x^5 - 2*x^6) / (1 - x)^9 + O(x^30))) \\ Colin Barker, Dec 13 2016

a(n) = (n^6 - 14*n^4 + 30*n^3 - 17*n^2 - 6*n + 6)*n^2/24 \\ Charles R Greathouse IV, Dec 13 2016

a(n) = (n^8 - 14*n^6 + 30*n^5 - 17*n^4 - 6*n^3 + 6*n^2)/24.
a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9).
G.f.: x^2*(1 + 81*x + 654*x^2 + 904*x^3 + 99*x^4 - 57*x^5 - 2*x^6) / (1 - x)^9. - Colin Barker, Dec 13 2016

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.

Original entry on oeis.org

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

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.

Original entry on oeis.org

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

Offset: 1

Heinrich Ludwig, Dec 22 2016

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.

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).

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

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

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 *)

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

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

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

A279442 Number of ways to place 8 points on an n X n square grid so that no more than 2 points are on a vertical or horizontal straight line.

Original entry on oeis.org

0, 0, 0, 90, 67950, 4531950, 109425330, 1460297160, 13112872920, 88456195800, 480149029800, 2196080372970, 8743233946590, 31033043111070, 99992483914050, 296626638016800, 819218054279520, 2125440234303840, 5218743585428640, 12201529135725450, 27304286810701950

Offset: 1

Heinrich Ludwig, Dec 22 2016

Column 9 of triangle A279445.

Rotations and reflections of placements are counted.

Heinrich Ludwig, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (17,-136,680,-2380,6188,-12376,19448,-24310,24310,-19448,12376,-6188,2380,-680,136,-17,1).

Cf. A279444, A279445, A197458.

Same problem but 2..7,9 points: A083374, A279437, A279438, A279439, A279440, A279441, A279443.

Table[n^2*(n - 1)^2*(n - 2)^2*(n - 3)^2*(n^8 + 12 n^7 - 54 n^6 - 444 n^5 + 1845 n^4 + 1392 n^3 - 11332 n^2 + 9660 n + 1260)/8!, {n, 21}] (* Michael De Vlieger, Dec 22 2016 *)

concat(vector(3), Vec(90*x^4*(1 +738*x +37656*x^2 +461802*x^3 +1893555*x^4 +2767824*x^5 +1058064*x^6 -331920*x^7 -140913*x^8 +15950*x^9 +3096*x^10 -90*x^11 -3*x^12) / (1 -x)^17 + O(x^30))) \\ Colin Barker, Dec 23 2016

a(n) = (n^16 -140*n^14 +756*n^13 +2506*n^12 -36540*n^11 +130940*n^10 -117432*n^9 -559615*n^8 +2186100*n^7 -3622360*n^6 +3228876*n^5 -1439892*n^4 +181440*n^3 +45360*n^2)/40320; factorized: a(n) = n^2*(n-1)^2*(n-2)^2*(n-3)^2*(n^8 +12*n^7 -54*n^6 -444*n^5 +1845*n^4 +1392*n^3 -11332*n^2 +9660*n +1260)/8!.
a(n) = SUM(1<=j<=17, C(17,j)*(-1)^(j-1)*a(n-j)).
G.f.: 90*x^4*(1 +738*x +37656*x^2 +461802*x^3 +1893555*x^4 +2767824*x^5 +1058064*x^6 -331920*x^7 -140913*x^8 +15950*x^9 +3096*x^10 -90*x^11 -3*x^12) / (1 -x)^17. - Colin Barker, Dec 23 2016

A279443 Number of ways to place 9 points on an n X n board so that no more than 2 points are on a vertical or horizontal straight line.

Original entry on oeis.org

0, 0, 0, 0, 22650, 4987800, 240023070, 5219088000, 68483325960, 630486309600, 4456523194200, 25647802519680, 125166919041450, 533442526857240, 2029603476250350, 7011735609715200, 22291042191643680, 65914292362262400, 182880685655641440, 479548000781222400

Offset: 1

Heinrich Ludwig, Dec 24 2016

Column 9 of triangle A279445.

Rotations and reflections of placements are counted.

Heinrich Ludwig, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (19,-171,969,-3876,11628, -27132,50388,-75582,92378,-92378,75582,-50388,27132,-11628,3876, -969,171,-19,1).

Cf. A279444, A279445, A197458.

Same problem but 2..8 points: A083374, A279437, A279438, A279439, A279440, A279441, A279442.

concat(vector(4), Vec(30*x^5*(755 +151915*x +4970934*x^2 +49653854*x^3 +187307071*x^4 +275138271*x^5 +119386656*x^6 -31251744*x^7 -19595619*x^8 +1706821*x^9 +667466*x^10 -26334*x^11 -2543*x^12 +17*x^13) / (1 -x)^19 + O(x^30))) \\ Colin Barker, Dec 24 2016

a(n) = (n^18 -204*n^16 +1260*n^15 +6846*n^14 -104076*n^13 +394504*n^12 +128520*n^11 -6237075*n^10 +24018372*n^9 -43820196*n^8 +30400020*n^7 +34251148*n^6 -99199296*n^5 +98504496*n^4 -47779200*n^3 +9434880*n^2)/362880; factorized: a(n) = n^2*(n-1)^2*(n-2)^2*(n-3)^2*(n-4)^2*(n^8 +20*n^7 +26*n^6 -820*n^5 -247*n^4 +9704*n^3 -9104*n^2 -14700*n +16380)/9!.
a(n) = SUM(1<=j<=19, C(19,j)*(-1)^(j-1)*a(n-j)).
G.f.: 30*x^5*(755 +151915*x +4970934*x^2 +49653854*x^3 +187307071*x^4 +275138271*x^5 +119386656*x^6 -31251744*x^7 -19595619*x^8 +1706821*x^9 +667466*x^10 -26334*x^11 -2543*x^12 +17*x^13) / (1 -x)^19. - Colin Barker, Dec 24 2016

A279450 Number of nonequivalent 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.

Original entry on oeis.org

0, 0, 2, 285, 9110, 126396, 1055025, 6266614, 29198740, 113262680, 380775248, 1140764611, 3108667306, 7824370092, 18407341855, 40855872764, 86201399496, 173952773328, 337453762782, 631982899545, 1146743732126, 2022212701212, 3474824082125, 5831439251154, 9576836632860

Offset: 1

Heinrich Ludwig, Dec 18 2016

Column 7 of A279453.
Rotations and reflections of placements are not counted. For numbers if they are to be counted see A279440.
For condition "no more than 2 points on straight lines at any angle", see A235457.

Heinrich Ludwig, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (6,-8,-22,69,-8,-176,168,182,-364,0,364,-182,-168,176,8,-69,22,8,-6,1).

Cf. A235457, A279440, A279452, A279453, A279454.

Same problem but 2,3,4,5,7 points: A014409, A279447, A279448, A279449, A279451.

concat(vector(2), Vec(x^3*(2 +273*x +7416*x^2 +74060*x^3 +375661*x^4 +1128403*x^5 +2194010*x^6 +2815082*x^7 +2424155*x^8 +1294751*x^9 +376028*x^10 -5296*x^11 -32173*x^12 -8195*x^13 +178*x^14 +122*x^15 +3*x^16) / ((1 -x)^13*(1 +x)^7) + O(x^40))) \\ Colin Barker, Dec 18 2016

a(n) = (n^12 - 55*n^10 + 210*n^9 + 93*n^8 - 2220*n^7 + 6052*n^6 - 8040*n^5 + 4236*n^4 + 3240*n^3 - 5872*n^2 + 2400*n)/5760 + IF(MOD(n, 2) = 1, 2*n^6 - 18*n^5 + 53*n^4 - 64*n^3 + 33*n^2 - 12*n + 5)/128.
a(n) = 6*a(n-1) - 8*a(n-2) - 22*a(n-3) + 69*a(n-4) - 8*a(n-5) - 176*a(n-6) + 168*a(n-7) + 182*a(n-8) - 364*a(n-9) + 364*a(n-11) - 182*a(n-12) - 168*a(n-13) + 176*a(n-14) + 8*a(n-15) - 69*a(n-16) + 22*a(n-17) + 8*a(n-18) - 6*a(n-19) + *a(n-20).
G.f.: x^3*(2 +273*x +7416*x^2 +74060*x^3 +375661*x^4 +1128403*x^5 +2194010*x^6 +2815082*x^7 +2424155*x^8 +1294751*x^9 +376028*x^10 -5296*x^11 -32173*x^12 -8195*x^13 +178*x^14 +122*x^15 +3*x^16) / ((1 -x)^13*(1 +x)^7). - Colin Barker, Dec 18 2016

cp's OEIS Frontend

A279445 Triangle read by rows: T(n, k) is the number of ways to place k 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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A279437 Number of ways to place 3 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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A279438 Number of ways to place 4 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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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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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.

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A279442 Number of ways to place 8 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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A279443 Number of ways to place 9 points on an n X n board so that no more than 2 points are on a vertical or horizontal straight line.

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A279450 Number of nonequivalent 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.

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