A194485 -id:A194485 - cp's OEIS frontend

A194481 Number of ways to arrange 4 indistinguishable points on an n X n X n triangular grid so that no four points are in the same row or diagonal.

0, 0, 15, 207, 1347, 5922, 20307, 58527, 148239, 339669, 718344, 1422564, 2666664, 4771221, 8201466, 13615266, 21922146, 34354926, 52555653, 78677613, 115505313, 166594428, 236433813, 330631785, 456128985, 621440235, 836927910, 1115109450

Offset: 1

R. H. Hardin, Aug 26 2011

nonn

Column 4 of A194485.

			Some solutions for 4 X 4 X 4:
.....0........0........1........0........0........0........0........0
....1.0......0.0......0.0......0.0......1.0......1.0......0.1......1.0
...1.1.1....1.0.1....1.0.1....1.0.1....1.0.0....1.0.0....0.0.1....1.0.1
..0.0.0.0..1.1.0.0..0.0.1.0..0.0.1.1..0.1.0.1..0.0.1.1..1.0.0.1..0.0.0.1

R. H. Hardin, Table of n, a(n) for n = 1..84

Cf. A194485.

Empirical: a(n) = (1/384)*n^8 + (1/96)*n^7 - (1/64)*n^6 - (13/120)*n^5 + (19/128)*n^4 + (7/96)*n^3 - (13/96)*n^2 + (1/40)*n.

Empirical g.f.: x^3*(5 + 24*x + 8*x^2 - 3*x^3 + x^4) / (1 - x)^9. - Colin Barker, May 05 2018

A194482 Number of ways to arrange 5 indistinguishable points on an n X n X n triangular grid so that no four points are in the same row or diagonal.

Original entry on oeis.org

0, 0, 6, 234, 2817, 19362, 94584, 365904, 1193283, 3413619, 8800704, 20845968, 46017972, 95710797, 189154056, 357631176, 650438802, 1143119610, 1948614426, 3232108278, 5230489803, 8277505236, 12835867968, 19537783320, 29235566685

Offset: 1

R. H. Hardin, Aug 26 2011

nonn

Column 5 of A194485.

			Some solutions for 4 X 4 X 4:
.....1........1........0........1........0........0........0........0
....1.1......1.0......1.1......0.0......0.1......1.1......1.0......1.1
...0.1.1....0.1.0....0.1.0....1.0.1....0.1.0....1.0.1....1.0.0....1.0.0
..0.0.0.0..0.1.0.1..0.1.1.0..1.1.0.0..0.1.1.1..0.1.0.0..1.1.0.1..1.1.0.0

R. H. Hardin, Table of n, a(n) for n = 1..46

Cf. A194485.

Empirical: a(n) = (1/3840)*n^10 + (1/768)*n^9 - (1/384)*n^8 - (59/1920)*n^7 + (281/3840)*n^6 + (149/3840)*n^5 - (5/24)*n^4 + (29/320)*n^3 + (11/80)*n^2 - (1/10)*n.

Empirical g.f.: 3*x^3*(2 + 56*x + 191*x^2 + 85*x^3 - 31*x^4 + 11*x^5 + x^6) / (1 - x)^11. - Colin Barker, May 05 2018

A194483 Number of ways to arrange 6 indistinguishable points on an n X n X n triangular grid so that no four points are in the same row or diagonal.

Original entry on oeis.org

0, 0, 1, 165, 4135, 47010, 337860, 1790472, 7622340, 27489825, 87018360, 247874770, 647091588, 1569661600, 3576049620, 7716906900, 15881735580, 31347485274, 59618165895, 109678780695, 195827638105, 340301983890, 576974687080

Offset: 1

R. H. Hardin, Aug 26 2011

nonn

Column 6 of A194485.

			Some solutions for 5 X 5 X 5:
......0..........1..........0..........0..........0..........0..........0
.....0.1........1.0........0.1........0.1........0.1........1.0........1.0
....0.1.0......0.1.1......0.0.1......1.0.1......0.1.0......0.1.0......0.1.1
...0.1.1.0....1.0.0.0....0.1.0.1....1.0.1.1....1.0.0.1....0.1.0.0....0.1.0.1
..0.0.1.1.0..0.0.1.0.0..1.0.0.1.0..0.0.0.0.0..1.0.0.0.1..0.0.1.1.1..1.0.0.0.0

R. H. Hardin, Table of n, a(n) for n = 1..29

Empirical: a(n) = (1/46080)*n^12 + (1/7680)*n^11 - (1/3072)*n^10 - (137/23040)*n^9 + (871/46080)*n^8 + (3107/161280)*n^7 - (5573/46080)*n^6 + (1157/23040)*n^5 + (2627/11520)*n^4 - (1121/5760)*n^3 - (181/1440)*n^2 + (11/84)*n

A194484 Number of ways to arrange 7 indistinguishable points on an n X n X n triangular grid so that no four points are in the same row or diagonal.

Original entry on oeis.org

0, 0, 0, 63, 4080, 83745, 927471, 6924357, 39196161, 180512640, 708150465, 2442836682, 7582054194, 21540941994, 56763356130, 140189208510, 327211061058, 726712057836, 1544399756262, 3155463833625, 6223010262480, 11886291766899

Offset: 1

R. H. Hardin, Aug 26 2011

nonn

Column 7 of A194485.

			Some solutions for 5 X 5 X 5:
......1..........0..........0..........1..........0..........0..........0
.....0.1........1.0........1.1........0.1........1.1........0.1........0.0
....1.1.1......0.1.0......1.1.1......1.0.0......0.0.0......0.1.1......1.0.1
...0.0.0.0....1.1.0.0....1.0.1.0....1.0.1.0....1.1.0.1....0.1.1.0....1.1.0.1
..1.1.0.0.0..0.1.0.1.1..0.0.0.0.0..0.1.0.0.1..0.1.0.1.0..1.0.0.0.1..1.0.0.1.0

Empirical: a(n) = (1/645120)*n^14 + (1/92160)*n^13 - (1/30720)*n^12 - (79/92160)*n^11 + (101/30720)*n^10 + (757/129024)*n^9 - (3049/92160)*n^8 - (34099/645120)*n^7 + (6613/15360)*n^6 - (16859/23040)*n^5 + (1043/3840)*n^4 + (2759/5040)*n^3 - (753/1120)*n^2 + (13/56)*n.

cp's OEIS Frontend

A194481 Number of ways to arrange 4 indistinguishable points on an n X n X n triangular grid so that no four points are in the same row or diagonal.

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A194482 Number of ways to arrange 5 indistinguishable points on an n X n X n triangular grid so that no four points are in the same row or diagonal.

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Examples

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Crossrefs

Formula

A194483 Number of ways to arrange 6 indistinguishable points on an n X n X n triangular grid so that no four points are in the same row or diagonal.

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Author

Keywords

Comments

Examples

Links

Formula

A194484 Number of ways to arrange 7 indistinguishable points on an n X n X n triangular grid so that no four points are in the same row or diagonal.

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Author

Keywords

Comments

Examples

Formula