A279453 -id:A279453 - 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.

A279452 Number of nonequivalent ways to place n 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

1, 2, 14, 202, 4361, 126396, 4444935, 182567862, 8560638040, 451107721000

Offset: 1

Heinrich Ludwig, Dec 23 2016

Rotations and reflections of placements are not counted. If they are to be counted, see A279444.

			There are a(3) = 14 nonequivalent ways to place 3 points on a 3 X 3 square grid under the condition mentioned above:
XX   XX.  XX.  XX.  XX.  XX.  X.X  X.X  X X  X..  X..  X..  .X.  .X.
..X  ...  X..  .X.  ...  ...  .X.  ...  ...  .XX  .X.  ..X  XX.  X.X
...  ..X  ...  ...  X..  .X.  ...  X..  .X.  ...  ..X  .X.  ...  ...

Cf. A279444, A279453.

A279447 Number of nonequivalent 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, 1, 14, 73, 301, 890, 2321, 5166, 10654, 20055, 35880, 60511, 98419, 153608, 233331, 343820, 496076, 699261, 969234, 1318885, 1770185, 2340646, 3059749, 3950618, 5051786, 6393075, 8023756, 9981531, 12328239, 15110740, 18405415, 22269656, 26796504, 32055353, 38158166

Offset: 1

Heinrich Ludwig, Dec 17 2016

Column 4 of A279453.
Rotations and reflections of placements are not counted. For numbers if they are to be counted see A279437.
For condition "no more than 2 points on straight lines at any angle", see A235454.

Heinrich Ludwig, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,1,-11,6,14,-14,-6,11,-1,-3,1).

Cf. A235454, A279437, A279452, A279453, A279454.

Same problem but 2, 4..7 points: A014409, A279448, A279449, A279450, A279451.

I:=[0,1,14,73,301,890,2321,5166,10654,20055,35880]; [n le 11 select I[n] else 3*Self(n-1)+Self(n-2)-11*Self(n-3)+ 6*Self(n-4)+14*Self(n-5)-14*Self(n-6)-6*Self(n-7)+11*Self(n-8)-Self(n-9)-3*Self(n-10)+Self(n-11): n in [1..40]]; // Vincenzo Librandi, Dec 17 2016

LinearRecurrence[{3, 1, -11, 6, 14, -14, -6, 11, -1, -3, 1},{0, 1, 14, 73, 301, 890, 2321, 5166, 10654, 20055, 35880}, 35] (* Vincenzo Librandi Dec 17 2016 *)

concat(0, Vec(x^2*(1 + 11*x + 30*x^2 + 79*x^3 + 62*x^4 + 55*x^5 + 4*x^6 - x^7 - x^8) / ((1 - x)^7*(1 + x)^4) + O(x^30))) \\ Colin Barker, Dec 17 2016

a(n) = (n^6 - 5*n^4 + 14*n^3 - 14*n^2 + 4*n)/48 + IF(MOD(n, 2) = 1, 2*n^3 - 3*n^2 + 1)/16.
a(n) = 3*a(n-1) + a(n-2) - 11*a(n-3) + 6*a(n-4) + 14*a(n-5) - 14*a(n-6) - 6*a(n-7) + 11*a(n-8) - a(n-9) - 3*a(n-10) + a(n-11).
G.f.: x^2*(1 + 11*x + 30*x^2 + 79*x^3 + 62*x^4 + 55*x^5 + 4*x^6 - x^7 - x^8) / ((1 - x)^7*(1 + x)^4). - Colin Barker, Dec 17 2016

A279448 Number of nonequivalent 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, 17, 202, 1397, 6582, 24185, 73496, 195086, 463875, 1013505, 2061426, 3956947, 7222992, 12640817, 21312992, 34801420, 55215621, 85424721, 129174250, 191397185, 278361226, 398108777, 560635032, 778491962, 1066995527, 1445034305, 1935301746, 2565356031, 3367870500

Offset: 1

Heinrich Ludwig, Dec 18 2016

Column 5 of A279453.
Rotations and reflections of placements are not counted. For numbers if they are to be counted see A279438.
For condition "no more than 2 points on straight lines at any angle", see A235455.

Heinrich Ludwig, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-1,-16,19,20,-45,0,45,-20,-19,16,1,-4,1).

Cf. A235455, A279438, A279452, A279453, A279454.

Same problem but 2,3,5,6,7 points: A014409, A279447, A279449, A279450, A279451.

concat(0, Vec(x^2*(1 +13*x +135*x^2 +622*x^3 +1449*x^4 +2143*x^5 +1557*x^6 +781*x^7 +34*x^8 -8*x^9 -8*x^10 +x^11) / ((1 -x)^9*(1 +x)^5) + O(x^40))) \\ Colin Barker, Dec 18 2016

a(n) = (n^8 - 14*n^6 + 30*n^5 + 12*n^4 - 60*n^3 + 40*n^2)/192 + IF(MOD(n, 2) = 1, 4*n^4 - 20*n^3 + 22*n^2 - 2*n - 7)/64.
a(n) = 4*a(n-1) - a(n-2) - 16*a(n-3) + 19*a(n-4) + 20*a(n-5) - 45*a(n-6) + 45*a(n-8) - 20*a(n-9) - 19*a(n-10) + 16*a(n-11) + a(n-12) - 4*a(n-13) + a(n-14).
G.f.: x^2*(1 +13*x +135*x^2 +622*x^3 +1449*x^4 +2143*x^5 +1557*x^6 +781*x^7 +34*x^8 -8*x^9 -8*x^10 +x^11) / ((1 -x)^9*(1 +x)^5). - Colin Barker, Dec 18 2016

A279449 Number of nonequivalent 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, 9, 306, 4361, 34059, 185181, 777280, 2710074, 8181558, 22067973, 54285858, 123791067, 264749849, 536146569, 1035584592, 1919530804, 3430908108, 5937810417, 9984193986, 16358592141, 26181281511, 41019234245, 63028246512, 95136210222, 141264963970, 206611069197

Offset: 1

Heinrich Ludwig, Dec 18 2016

Column 6 of A279453.
Rotations and reflections of placements are not counted. For numbers if they are to be counted see A279439.
For condition "no more than 2 points on straight lines at any angle", see A235456.

Heinrich Ludwig, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5,-4,-20,40,16,-100,44,110,-110,-44,100,-16,-40,20,4,-5,1).

Cf. A235456, A279439, A279452, A279453, A279454.

Same problem but 2,3,4,6,7 points: A014409, A279447, A279448, A279450, A279451.

concat(vector(2), Vec(x^3*(9 +261*x +2867*x^2 +13658*x^3 +38090*x^4 +62447*x^5 +67142*x^6 +41996*x^7 +15541*x^8 +955*x^9 -761*x^10 -278*x^11 -8*x^12 +x^13) / ((1 -x)^11*(1 +x)^6) + O(x^40))) \\ Colin Barker, Dec 18 2016

a(n) = (n^10 - 30*n^8 + 90*n^7 - 27*n^6 - 218*n^5 + 340*n^4 - 340*n^3 + 376*n^2 - 192*n)/960 + IF(MOD(n, 2) = 1, 2*n^5 - 9*n^4 + 14*n^3 - 6*n^2 - 4*n + 3)/64.
a(n) = 5*a(n-1) - 4*a(n-2) - 20*a(n-3) + 40*a(n-4) + 16*a(n-5) - 100*a(n-6) + 44*a(n-7) + 110*a(n-8) - 110*a(n-9) - 44*a(n-10) + 100*a(n-11) - 16*a(n-12) - 40*a(n-13) + 20*a(n-14) + 4*a(n-15) - 5*a(n-16) + a(n-17).
G.f.: x^3*(9 +261*x +2867*x^2 +13658*x^3 +38090*x^4 +62447*x^5 +67142*x^6 +41996*x^7 +15541*x^8 +955*x^9 -761*x^10 -278*x^11 -8*x^12 +x^13) / ((1 -x)^11*(1 +x)^6). - Colin Barker, Dec 18 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

A279451 Number of nonequivalent 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, 115, 11810, 326190, 4444935, 38675954, 246563232, 1248782460, 5296300670, 19499431941, 63958228738, 190528987506, 523151460045, 1339408935540, 3227223506896, 7372750196952, 16069268866908, 33586411339335, 67610793877650, 131569779776182, 248290280743571

Offset: 1

Heinrich Ludwig, Dec 22 2016

Column 8 of A279453.
Rotations and reflections of placements are not counted. For numbers if they are to be counted see A279441.
For condition "no more than 2 points on straight lines at any angle", see A235458.

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

Cf. A235458, A279441, A279452, A279453, A279454.

Same problem but 2..6 points: A014409, A279447, A279448, A279449, A279450.

Table[(n^14 - 91 n^12 + 420 n^11 + 693 n^10 - 10500 n^9 + 33647 n^8 - 45316 n^7 + 3682 n^6 + 62300 n^5 - 51996 n^4 - 28504 n^3 + 54384 n^2 - 18720 n)/40320 + Boole[OddQ@ n] (2 n^7 - 17 n^6 + 50 n^5 - 59 n^4 + 38 n^3 - 71 n^2 + 102 n - 45)/384, {n, 23}] (* or *)
Rest@ CoefficientList[Series[x^4*(115 + 11005 x + 245015 x^2 + 2317550 x^3 + 12037814 x^4 + 39232894 x^5 + 85494738 x^6 + 129182670 x^7 + 135873108 x^8 + 97856368 x^9 + 44499480 x^10 + 9709722 x^11 - 1359254 x^12 - 1352974 x^13 - 257282 x^14 + 13866 x^15 + 7705 x^16 + 419 x^17 + x^18)/((1 - x)^15*(1 + x)^8), {x, 0, 23}], x] (* Michael De Vlieger, Dec 23 2016 *)

concat(vector(3), Vec(x^4*(115 +11005*x +245015*x^2 +2317550*x^3 +12037814*x^4 +39232894*x^5 +85494738*x^6 +129182670*x^7 +135873108*x^8 +97856368*x^9 +44499480*x^10 +9709722*x^11 -1359254*x^12 -1352974*x^13 -257282*x^14 +13866*x^15 +7705*x^16 +419*x^17 +x^18) / ((1 -x)^15*(1 +x)^8) + O(x^30))) \\ Colin Barker, Dec 23 2016

a(n) = (n^14 -91*n^12 +420*n^11 +693*n^10 -10500*n^9 +33647*n^8 -45316*n^7 +3682*n^6 +62300*n^5 -51996*n^4 -28504*n^3 +54384*n^2 -18720*n)/40320 + IF(MOD(n, 2) = 1, 2*n^7 -17*n^6 +50*n^5 -59*n^4 +38*n^3 -71*n^2 +102*n -45)/384.

G.f.: x^4*(115 +11005*x +245015*x^2 +2317550*x^3 +12037814*x^4 +39232894*x^5 +85494738*x^6 +129182670*x^7 +135873108*x^8 +97856368*x^9 +44499480*x^10 +9709722*x^11 -1359254*x^12 -1352974*x^13 -257282*x^14 +13866*x^15 +7705*x^16 +419*x^17 +x^18) / ((1 -x)^15*(1 +x)^8). - Colin Barker, Dec 23 2016

A279454 Number of nonequivalent ways to place k>=0 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

2, 6, 54, 1026, 38931, 2198241, 167110761, 16248457762, 1960813221768

Offset: 1

Heinrich Ludwig, Dec 24 2016

Row sum of triangle A279453.

Rotations and reflections of a placement are not counted. For numbers if they are to be counted, see A197458.

Cf. A197458, A279453.

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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A279452 Number of nonequivalent ways to place n 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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A279447 Number of nonequivalent 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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A279448 Number of nonequivalent 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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A279449 Number of nonequivalent 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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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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A279451 Number of nonequivalent 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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A279454 Number of nonequivalent ways to place k>=0 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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