A279452 -id:A279452 - cp's OEIS frontend

A279444 Number of 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.

1, 6, 78, 1428, 34020, 1005720, 35526120, 1460297160, 68483325960, 3608847088560, 211125189789360, 13577790144108960, 952129825291925280, 72303731169467878080, 5911241852357814772800, 517670861532464104264800, 48347072773751574327304800

Offset: 1

Heinrich Ludwig, Dec 23 2016

nonn

Rotations and reflections of placements are counted. If they are to be ignored, see A279452.

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..100

Cf. A279452, A279445.

a(11)-a(17) from Hiroaki Yamanouchi, Jan 04 2017

A279453 Triangle read by rows: T(n, k) is the number of nonequivalent 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.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 3, 8, 14, 17, 9, 2, 1, 3, 21, 73, 202, 306, 285, 115, 20, 1, 6, 49, 301, 1397, 4361, 9110, 11810, 8679, 2929, 288, 1, 6, 93, 890, 6582, 34059, 126396, 326190, 568134, 624875, 390426, 111798, 8791, 1, 10, 171, 2321, 24185, 185181, 1055025

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 that can be placed under the condition mentioned.
Rotations and reflections of placements are not counted. If they are to be counted, see A279445.
For condition "no more than 2 points on a straight line at any angle", see A235453.

			The table begins with T(1, 0):
1 1
1 1  2   1    1
1 3  8  14   17    9    2
1 3 21  73  202  306  285   115   20
1 6 49 301 1397 4361 9110 11810 8679 2929 288
...
T(4, 3) = 73 because there are 73 nonequivalent ways to place 3 points on a 4 X 4 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 A279454.
Columns 2..8: A008805, A014409, A279454, A279455, A279456, A279457, A279458.
Diagonal T(n, n) is A279452.
Cf. A279445, A235453.

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

cp's OEIS Frontend

A279444 Number of 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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A279453 Triangle read by rows: T(n, k) is the number of nonequivalent 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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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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