A194480 -id:A194480 - cp's OEIS frontend

A194475 Number of ways to arrange 3 indistinguishable points on an n X n X n triangular grid so that no three points are in the same row or diagonal.

0, 1, 17, 105, 410, 1225, 3066, 6762, 13560, 25245, 44275, 73931, 118482, 183365, 275380, 402900, 576096, 807177, 1110645, 1503565, 2005850, 2640561, 3434222, 4417150, 5623800, 7093125, 8868951, 11000367, 13542130, 16555085, 20106600

Offset: 1

R. H. Hardin, Aug 26 2011

nonn

Column 3 of A194480.

			The 17 solutions for 3 X 3 X 3:
.
    1         1         1         1         1         1
   1 1       1 0       1 0       0 1       0 1       0 0
  0 0 0     0 1 0     0 0 1     1 0 0     0 1 0     1 1 0
         1         1         0         0         0
        0 0       0 0       1 1       1 1       1 1
       1 0 1     0 1 1     1 0 0     0 1 0     0 0 1
    0         0         0         0         0         0
   1 0       1 0       1 0       0 1       0 1       0 1
  1 1 0     1 0 1     0 1 1     1 1 0     1 0 1     0 1 1
[edited by _Jon E. Schoenfield_, May 05 2018]

R. H. Hardin, Table of n, a(n) for n = 1..200

Cf. A194480.

Empirical: a(n) = (1/48)*n^6 + (1/16)*n^5 - (3/16)*n^4 + (1/48)*n^3 + (1/6)*n^2 - (1/12)*n.

Empirical g.f.: x^2*(1 + 10*x + 7*x^2 - 3*x^3) / (1 - x)^7. - Colin Barker, May 05 2018

A194476 Number of ways to arrange 4 indistinguishable points on an n X n X n triangular grid so that no three points are in the same row or diagonal.

Original entry on oeis.org

0, 0, 6, 114, 879, 4284, 15729, 47565, 124803, 293733, 634293, 1277133, 2426424, 4389567, 7615062, 12739902, 20647962, 32540958, 50023656, 75205116, 110817861, 160356966, 228241167, 319998195, 442476645, 604086795, 815072895, 1087819551

Offset: 1

R. H. Hardin, Aug 26 2011

nonn

Column 4 of A194480.

			All solutions for 3 X 3 X 3:
....0......1......0......1......1......0
...1.1....0.1....1.1....1.1....1.0....1.1
..0.1.1..1.1.0..1.1.0..0.1.0..0.1.1..1.0.1

R. H. Hardin, Table of n, a(n) for n = 1..112

Cf. A194480

Empirical: a(n) = (1/384)*n^8 + (1/96)*n^7 - (5/64)*n^6 + (13/240)*n^5 + (27/128)*n^4 - (23/96)*n^3 - (13/96)*n^2 + (7/40)*n.

Empirical g.f.: x^3*(2 + 20*x + 23*x^2 - 9*x^3 - x^4) / (1 - x)^9. - Colin Barker, May 05 2018

A194477 Number of ways to arrange 5 indistinguishable points on an n X n X n triangular grid so that no three points are in the same row or diagonal.

Original entry on oeis.org

0, 0, 0, 39, 909, 8568, 50526, 221508, 789453, 2412333, 6542316, 16127397, 36762726, 78495417, 158548572, 305303544, 563965038, 1004432454, 1732013856, 2901747051, 4737236427, 7555075374, 11796103242, 18064943820, 27179490195, 40232239515

Offset: 1

R. H. Hardin, Aug 26 2011

nonn

Column 5 of A194480.

			Some solutions for 4 X 4 X 4:
.....0........0........0........0........0........1........1........0
....0.1......0.1......1.0......1.0......1.0......0.0......1.0......0.1
...1.1.0....1.0.1....1.0.1....0.1.1....1.0.1....1.0.1....0.1.0....1.0.1
..0.0.1.1..0.1.1.0..0.1.1.0..1.1.0.0..0.1.0.1..0.1.1.0..0.1.0.1..1.1.0.0

R. H. Hardin, Table of n, a(n) for n = 1..47

Cf. A194480.

Empirical: a(n) = (1/3840)*n^10 + (1/768)*n^9 - (7/384)*n^8 + (37/1920)*n^7 + (737/3840)*n^6 - (2347/3840)*n^5 + (101/192)*n^4 + (93/320)*n^3 - (7/10)*n^2 + (3/10)*n.

Empirical g.f.: 3*x^4*(13 + 160*x + 238*x^2 - 54*x^3 - 51*x^4 + 9*x^5) / (1 - x)^11. - Colin Barker, May 05 2018

A194478 Number of ways to arrange 6 indistinguishable points on an n X n X n triangular grid so that no three points are in the same row or diagonal.

Original entry on oeis.org

0, 0, 0, 1, 337, 8733, 96478, 668028, 3413828, 14054915, 49171641, 151422970, 420674150, 1073422309, 2550004472, 5699074284, 12082541388, 24462528078, 47555986746, 89173692795, 161899772067, 285517344145, 490447009030

Offset: 1

R. H. Hardin, Aug 26 2011

nonn

			Some solutions for 5 X 5 X 5:
      0          0          1          0          0          1          1
     0 0        1 0        0 0        1 1        1 0        0 0        0 1
    1 1 0      0 0 1      0 1 0      0 0 0      1 0 1      0 1 0      1 0 0
   0 0 1 1    0 1 0 1    0 1 1 0    1 0 0 1    0 1 0 1    0 1 1 0    0 0 1 0
  1 0 0 0 1  1 0 0 1 0  1 0 0 0 1  0 1 1 0 0  0 1 0 0 0  0 0 1 0 1  0 0 1 1 0

Manuel Kauers and Christoph Koutschan, Table of n, a(n) for n = 1..1000 (terms 1..31 from R. H. Hardin).
M. Kauers and C. Koutschan, Some D-finite and some possibly D-finite sequences in the OEIS, arXiv:2303.02793 [cs.SC], 2023.

Column 6 of A194480.

From Manuel Kauers and Christoph Koutschan, Mar 02 2023: (Start)
a(n) = (1/256)*(-1)^n*(2*n - 7)*(n^2 - 7*n + 13) + (1/322560)*(7*n^12 + 42*n^11 - 945*n^10 + 1274*n^9 + 26089*n^8 - 128810*n^7 + 175693*n^6 + 205366*n^5 - 810796*n^4 + 601328*n^3 + 354172*n^2 - 582180*n + 114660).
Recurrence: (n-2)*(14*n^11 + 70*n^10 - 2051*n^9 + 5299*n^8 + 50106*n^7 - 359946*n^6 + 953463*n^5 - 1085555*n^4 - 364412*n^3 + 3593716*n^2 - 6028304*n + 3620736)*a(n+2) + (-126*n^11 - 966*n^10 + 13377*n^9 + 4662*n^8 - 354550*n^7 + 1123664*n^6 - 1113309*n^5 + 85056*n^4 + 1719696*n^3 - 7286000*n^2 + 10210192*n - 3854400)*a(n+1) - (n+2)*(14*n^11 + 224*n^10 - 581*n^9 - 7700*n^8 + 31682*n^7 - 11948*n^6 - 91561*n^5 + 168104*n^4 - 482042*n^3 + 1253272*n^2 - 1293160*n + 383136)*a(n) = 0. (End)

A194479 Number of ways to arrange 7 indistinguishable points on an n X n X n triangular grid so that no three points are in the same row or diagonal.

Original entry on oeis.org

0, 0, 0, 0, 15, 3525, 98473, 1237434, 9821400, 57367112, 268378248, 1059987987, 3661533037, 11341971885, 32090198922, 84099053568, 206386405560, 478423135296, 1054963957500, 2225704950054, 4514320792971, 8838377453829

Offset: 1

R. H. Hardin, Aug 26 2011

nonn

Column 7 of A194480.

			Some solutions for 5 X 5 X 5:
......0..........0..........1..........0..........0..........1..........1
.....1.0........0.1........0.0........1.0........0.1........0.0........0.0
....0.1.1......1.1.0......1.0.1......1.0.1......1.0.1......1.1.0......0.1.1
...1.0.0.1....1.0.0.1....0.1.1.0....0.1.1.0....1.0.1.0....0.0.1.1....1.1.0.0
..0.1.1.0.0..0.0.1.1.0..0.1.0.1.0..0.1.0.0.1..0.1.0.1.0..0.1.1.0.0..0.0.1.1.0

R. H. Hardin, Table of n, a(n) for n = 1..24

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A194475 Number of ways to arrange 3 indistinguishable points on an n X n X n triangular grid so that no three points are in the same row or diagonal.

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A194476 Number of ways to arrange 4 indistinguishable points on an n X n X n triangular grid so that no three points are in the same row or diagonal.

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A194477 Number of ways to arrange 5 indistinguishable points on an n X n X n triangular grid so that no three points are in the same row or diagonal.

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A194478 Number of ways to arrange 6 indistinguishable points on an n X n X n triangular grid so that no three points are in the same row or diagonal.

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A194479 Number of ways to arrange 7 indistinguishable points on an n X n X n triangular grid so that no three points are in the same row or diagonal.

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Author

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