A239567 -id:A239567 - cp's OEIS frontend

A239572 Triangle T(n, k) = Numbers of non-equivalent (mod D_3) ways to place k points on a triangular grid of side n so that no two of them are adjacent. Triangle read by rows.

1, 1, 2, 2, 1, 3, 6, 6, 1, 4, 16, 32, 24, 7, 1, 5, 32, 113, 200, 176, 66, 6, 7, 60, 329, 1053, 1976, 2096, 1162, 302, 34, 2, 8, 100, 790, 3932, 12565, 25676, 32963, 25638, 11294, 2493, 222, 7, 10, 160, 1702, 11988, 57275, 187984, 425329, 658608, 684671, 462519

Offset: 1

Heinrich Ludwig, Mar 22 2014

Triangle T(n, k) is irregularly shaped: 1 <= k <= A239438(n). First row corresponds to n = 1.
The maximal number of points that can be placed on a triangular grid of side n so that no two of them are adjacent is given by A239438(n).
Without the restriction "non-equivalent (mod D_3)" numbers are given by A239567.

			Triangle begins
  1;
  1;
  2,   2,   1;
  3,   6,   6,    1;
  4,  16,  32,   24,     7,     1;
  5,  32, 113,  200,   176,    66,     6;
  7,  60, 329, 1053,  1976,  2096,  1162,   302,    34,    2;
  8, 100, 790, 3932, 12565, 25676, 32963, 25638, 11294, 2493, 222, 7;

Heinrich Ludwig, Table of n, a(n) for n = 1..136

Cf. A239438, A239567.
Column 1 is A001399,
Column 2 is A032091,
Column 3 is A239573,
Column 4 is A239574,
Column 5 is A239575,
Column 6 is A279446.

A239568 Number of ways to place 2 points on a triangular grid of side n so that they are not adjacent.

Original entry on oeis.org

0, 6, 27, 75, 165, 315, 546, 882, 1350, 1980, 2805, 3861, 5187, 6825, 8820, 11220, 14076, 17442, 21375, 25935, 31185, 37191, 44022, 51750, 60450, 70200, 81081, 93177, 106575, 121365, 137640, 155496, 175032, 196350, 219555, 244755, 272061, 301587, 333450, 367770

Offset: 2

Heinrich Ludwig, Mar 22 2014

Heinrich Ludwig, Table of n, a(n) for n = 2..1000
M. J. Hay, J. Schiff, N. J. Fisch, Maximal energy extraction under discrete diffusive exchange, arXiv preprint arXiv:1508.03499, 2015
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1)

Cf. A239567, A032091, A239569 (3 points), A239570 (4 points), A239571 (5 points), A282998 (6 points).

Regarding the third formula, see similar sequences listed in A241765.

concat(0, Vec(3*x^3*(x-2)/(x-1)^5 + O(x^100))) \\ Colin Barker, Mar 22 2014

a(n) = n*(n-1)*(n-2)*(n+5)/8.
G.f.: 3*x^3*(x-2) / (x-1)^5. - Colin Barker, Mar 22 2014
a(n) = Sum_{i=0..n} (i+5)*A000217(i). - Bruno Berselli, Apr 29 2014
a(n) = t(t(n,k),n) + n, where t(n,k) = n*(n+1)/2 + k*n and t(n,1) = A000096(n). - Bruno Berselli, Feb 28 2017

A239569 Number of ways to place 3 points on a triangular grid of side n so that no two of them are adjacent.

Original entry on oeis.org

0, 1, 21, 151, 615, 1845, 4571, 9926, 19566, 35805, 61765, 101541, 160381, 244881, 363195, 525260, 743036, 1030761, 1405221, 1886035, 2495955, 3261181, 4211691, 5381586, 6809450, 8538725, 10618101, 13101921, 16050601, 19531065, 23617195, 28390296, 33939576

Offset: 2

Heinrich Ludwig, Mar 22 2014

Rotations and reflections of placements are counted. If they are to be ignored, see A239573.

Vincenzo Librandi, Table of n, a(n) for n = 2..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1)

Cf. A239567, A239573, A239568 (2 points), A239570 (4 points), A239571 (5 points), A282998 (6 points).

[(n^2-3*n+2)*(n^4+6*n^3-23*n^2-92*n+264)/48: n in [2..40]]; // Vincenzo Librandi, Mar 23 2014

CoefficientList[Series[- x (11 x^4 - 36 x^3 + 25 x^2 + 14 x + 1)/(x - 1)^7, {x, 0, 40}], x] (* Vincenzo Librandi, Mar 23 2014 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,1,21,151,615,1845,4571},50] (* Harvey P. Dale, Aug 08 2023 *)

concat(0, Vec(-x^3*(11*x^4-36*x^3+25*x^2+14*x+1)/(x-1)^7 + O(x^100))) \\ Colin Barker, Mar 22 2014

a(n) = (n-1)*(n-2)*(n^4+6*n^3-23*n^2-92*n+264)/48.

G.f.: -x^3*(11*x^4-36*x^3+25*x^2+14*x+1) / (x-1)^7. - Colin Barker, Mar 22 2014

A239570 Number of ways to place 4 points on a triangular grid of side n so that no two of them are adjacent.

Original entry on oeis.org

0, 1, 114, 1137, 6100, 23265, 71211, 186739, 436437, 932850, 1856305, 3483546, 6224439, 10668112, 17640000, 28271370, 44083006, 67084839, 99893412, 145869175, 209275710, 295463091, 411077689, 564300837, 765118875, 1025627200, 1360371051, 1786725864, 2325320137

Offset: 3

Heinrich Ludwig, Mar 22 2014

Rotations and reflections of placements are counted. If they are to be ignored see A239574.

Vincenzo Librandi, Table of n, a(n) for n = 3..1000
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1)

Cf. A239567, A239574, A239568 (2 points), A239569 (3 points), A239571 (5 points), A282998 (6 points).

[(n^2-5*n+6)*(n^6+9*n^5-39*n^4-353*n^3+950*n^2 +4040*n-11904)/384: n in [3..40]]; // Vincenzo Librandi, Mar 23 2014

CoefficientList[Series[x (38 x^6 - 156 x^5 + 153 x^4 + 113 x^3 - 147 x^2 - 105 x - 1)/(x - 1)^9, {x, 0, 40}], x] (* Vincenzo Librandi, Mar 23 2014 *)
LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{0,1,114,1137,6100,23265,71211,186739,436437},30] (* Harvey P. Dale, May 28 2025 *)

concat(0, Vec(x^4*(38*x^6-156*x^5+153*x^4+113*x^3-147*x^2-105*x-1)/(x-1)^9 + O(x^100))) \\ Colin Barker, Mar 22 2014

a(n) = (n-2)*(n-3)*(n^6+9*n^5-39*n^4-353*n^3+950*n^2+4040*n-11904)/384.

G.f.: x^4*(38*x^6-156*x^5+153*x^4+113*x^3-147*x^2-105*x-1) / (x-1)^9. - Colin Barker, Mar 22 2014

A239571 Number of ways to place 5 points on a triangular grid of side n so that no two of them are adjacent.

Original entry on oeis.org

0, 0, 27, 999, 11565, 74811, 342042, 1239525, 3799488, 10259640, 25076952, 56552364, 119324403, 238062357, 452774595, 826245798, 1454229216, 2479147536, 4108199481, 6636929805, 10479498849, 16207085223, 24596072424, 36687908235, 53862785520, 77929575480

Offset: 3

Heinrich Ludwig, Mar 22 2014

Rotations and reflections of placements are counted. If they are to be ignored see A239575.

Vincenzo Librandi, Table of n, a(n) for n = 3..1000
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1)

Cf. A239567, A239575, A239568 (2 points), A239569 (3 points), A239570 (4 points), A282998 (6 points).

[(n^2-7*n+12)*(n^8+12*n^7-58*n^6-860*n^5+2141*n^4 +23728*n^3-61316*n^2-244928*n+770880)/3840: n in [3..40]]; // Vincenzo Librandi, Mar 23 2014

CoefficientList[Series[- 3 x^2 (40 x^8 - 185 x^7 + 198 x^6 + 213 x^5 - 243 x^4 - 638 x^3 + 687 x^2 + 234 x + 9)/(x - 1)^11, {x, 0, 40}], x] (* Vincenzo Librandi, Mar 23 2014 *)

concat([0,0], Vec(-3*x^5*(40*x^8-185*x^7+198*x^6+213*x^5-243*x^4-638*x^3+687*x^2+234*x+9)/(x-1)^11 + O(x^100))) \\ Colin Barker, Mar 22 2014

a(n) = (n -3) * (n -4) * (n^8 +12*n^7 -58*n^6 -860*n^5 +2141*n^4 +23728*n^3 -61316*n^2 -244928*n +770880)/3840.

G.f.: -3*x^5*(40*x^8-185*x^7+198*x^6+213*x^5-243*x^4-638*x^3+687*x^2+234*x+9) / (x-1)^11. - Colin Barker, Mar 22 2014

A282998 Number of ways to place 6 points on a triangular grid of side n so that no two of them are adjacent.

Original entry on oeis.org

0, 0, 1, 353, 12231, 153194, 1124820, 5893221, 24425212, 85152341, 259805430, 712840480, 1793423456, 4197531636, 9240962666, 19301854131, 38514786780, 73828909906, 136581190475, 244784427831, 426389859697, 723857976770, 1200460734396, 1948846090829, 3102524331336

Offset: 3

Heinrich Ludwig, Feb 26 2017

Rotations and reflections of placements are counted. If they are to be ignored, see A279446.

			There is a(5) = 1 way to place 6 points on a triangular grid of side n = 5:
        X
       . .
      X . X
     . . . .
    X . X . X

Heinrich Ludwig, Table of n, a(n) for n = 3..1000
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Cf. A279446, A239567, A239568 (2 points), A239569 (3 points), A239570 (4 points), A239571 (5 points).

A282998:=n->(n^12 + 6*n^11 - 195*n^10 - 670*n^9 + 17455*n^8 + 13426*n^7 - 836249*n^6 + 1252990*n^5 + 19599884*n^4 - 68542552*n^3 - 131400416*n^2 + 974223360*n - 1308856320)/46080: 0,seq(A282998(n), n=4..30); # Wesley Ivan Hurt, Apr 10 2017

Drop[CoefficientList[Series[(x^5 * (1 + 340 * x + 7720 * x^2 + 21439 * x^3 - 12927 * x^4 - 27265 * x^5 + 28385 * x^6 - 6252 * x^7 - 116 * x^8 - 2365 * x^9 + 1787 * x^10 - 352 * x^11) / (1 - x)^13 ),{x,0,27}],x],3] (* Indranil Ghosh, Feb 26 2017, from the g.f. by Colin Barker *)

concat(vector(2), Vec(x^5*(1 + 340*x + 7720*x^2 + 21439*x^3 - 12927*x^4 - 27265*x^5 + 28385*x^6 - 6252*x^7 - 116*x^8 - 2365*x^9 + 1787*x^10 - 352*x^11) / (1 - x)^13 + O(x^30))) \\ Colin Barker, Feb 26 2017

a(n) = (n^12 + 6*n^11 - 195*n^10 - 670*n^9 + 17455*n^8 + 13426*n^7 - 836249*n^6 + 1252990*n^5 + 19599884*n^4 - 68542552*n^3 - 131400416*n^2 + 974223360*n - 1308856320)/46080 for n>=4.

G.f.: x^5*(1 + 340*x + 7720*x^2 + 21439*x^3 - 12927*x^4 - 27265*x^5 + 28385*x^6 - 6252*x^7 - 116*x^8 - 2365*x^9 + 1787*x^10 - 352*x^11) / (1 - x)^13. - Colin Barker, Feb 26 2017

A239438 Maximal number of points that can be placed on a triangular grid of side n so that there is no pair of adjacent points.

Original entry on oeis.org

1, 1, 3, 4, 6, 7, 10, 12, 15, 19, 22, 26, 31, 35, 40, 46, 51, 57, 64, 70, 77, 85, 92, 100, 109, 117, 126, 136, 145, 155, 166, 176, 187, 199, 210, 222, 235, 247, 260, 274, 287, 301, 316, 330, 345, 361, 376, 392, 409

Offset: 1

Heinrich Ludwig, Mar 18 2014

In other words, the independence number of the (n-1)-triangular grid graph.
Apart from a(3) and a(5) same as A007997(n+4) and A058212(n+2). - Eric W. Weisstein, Jun 14 2017
Also the independence number of the n-triangular honeycomb king graph. - Eric W. Weisstein, Sep 06 2017

			On a triangular grid of side 5 at most a(5) = 6 points (X) can be placed so that there is no pair of adjacent points.
      X
     . .
    X . X
   . . . .
  X . X . X

Colin Barker, Table of n, a(n) for n = 1..1000
A. V. Geramita, D. Gregory, and L. Roberts, Monomial ideals and points in projective space, J. Pure Applied Alg 40 (1986), pp. 33-62.
Stan Wagon, Graph Theory Problems from Hexagonal and Traditional Chess, The College Mathematics Journal, Vol. 45, No. 4, September 2014, pp. 278-287.
Eric Weisstein's World of Mathematics, Independence Number
Eric Weisstein's World of Mathematics, Triangular Grid Graph
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Cf. A007997, A058212, A239567.

Table[1/18 (Piecewise[{{28, n == 2 || n == 4}}, 10] + 3 n (3 + n) + 8 Cos[(2 n Pi)/3]), {n, 0, 20}] (* Eric W. Weisstein, Jun 14 2017 *)

Vec(x*(x^9-2*x^8+2*x^7-3*x^6+3*x^5-2*x^4+2*x^3-2*x^2+x-1)/((x-1)^3*(x^2+x+1)) + O(x^100)) \\ Colin Barker, Feb 08 2015

a(n) = ceiling(n(n+1)/6) for n > 5, see Geramita, Gregory, & Roberts theorem 5.4. - Charles R Greathouse IV, Dec 04 2014

G.f.: x*(x^9-2*x^8+2*x^7-3*x^6+3*x^5-2*x^4+2*x^3-2*x^2+x-1) / ((x-1)^3*(x^2+x+1)). - Colin Barker, Feb 08 2015

Extended by Charles R Greathouse IV, Dec 04 2014

A243211 Triangle T(n, k) = Numbers of ways to place k points on a triangular grid of side n so that no three of them are vertices of an equilateral triangle with sides parallel to the grid. Triangle read by rows.

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 6, 15, 15, 3, 1, 10, 45, 107, 128, 63, 10, 1, 15, 105, 428, 1062, 1566, 1276, 507, 69, 1, 21, 210, 1282, 5160, 13971, 25191, 29235, 20508, 7747, 1251, 42, 1, 1, 28, 378, 3198, 18591, 77124, 231090, 498097, 759117, 792942, 540361, 222597, 49053

Offset: 1

Heinrich Ludwig, Jun 09 2014

The triangle T(n, k) is irregularly shaped: 0 <= k <= A227308(n). First row corresponds to n = 1.

The maximal number of points that can be placed on a triangular grid of side n so that no three of them form an equilateral triangle with sides parallel to the grid is given by A227308(n).

			The triangle begins:
  1,  1;
  1,  3,   3;
  1,  6,  15,   15,    3;
  1, 10,  45,  107,  128,    63,    10,
  1, 15, 105,  428, 1062,  1566,  1276,   507,    69,
  1, 21, 210, 1282, 5160, 13971, 25191, 29235, 20508, 7747, 1251, 42, 1;
  ...
There is T(6, 12) = 1 way to place 12 points (x) on the grid obeying the rule in the definition of the sequence:
           .
          x x
         x . x
        x . . x
       x . . . x
      . x x x x .

Heinrich Ludwig, Table of n, a(n) for n = 1..165

Cf. A227308, A243207, A084546, A234251, A239567, A240439, A194136, A000217 (column 2), A050534 (column 3), A243212 (column 4), A243213 (column 5), A243214 (column 6).

A027740 Number of independent subsets of nodes in graph formed from n-fold subdivision of triangle.

Original entry on oeis.org

1, 2, 4, 14, 60, 384, 3318, 40638, 689636, 16383974, 542420394, 25075022590, 1617185558560, 145563089994148, 18283036276489970, 3204638749437865046, 783848125594781710150, 267554112823378352976752

Offset: 0

R. H. Hardin

In other words, number of independent vertex sets (and vertex covers) in the (n-1)-triangular grid graph. - Eric W. Weisstein, Jun 14 2017

Number of planar n X n X n binary triangular grids with no more than 1 one in any similarly oriented 2 X 2 X 2 subtriangle. - R. H. Hardin, Dec 27 2008

Liang Kai, Table of n, a(n) for n = 0..40
Kai Liang, Independent Set Enumeration and Estimation of Related Constants of Grid Graphs, arXiv:2507.04007 [math.CO], 2025. See pp. 1, 10.
Eric Weisstein's World of Mathematics, Independent Vertex Set
Eric Weisstein's World of Mathematics, Triangular Grid Graph
Eric Weisstein's World of Mathematics, Vertex Cover

Cf. A239567, A239438, A287064.

A358532 a(n) is the row position of the next open point in the structure generated by adding the largest diamond possible at the next open point on a triangular grid of side n. See Comments and Example sections for more details.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 4, 1, 3, 7, 1, 3, 6, 4, 10, 1, 9, 4, 7, 9, 5, 14, 1, 11, 5, 7, 8, 11, 14, 19, 1, 6, 6, 24, 9, 14, 20, 1, 8, 8, 8, 20, 8, 19, 24, 30, 15, 19, 19, 19, 27, 1, 19, 15, 16, 20, 28, 8, 39, 11, 24, 1, 11, 16, 26, 28, 29, 30, 39, 50, 20, 31, 32, 33

Offset: 1

John Tyler Rascoe, Nov 20 2022

A structure of diamonds is built up successively by adding the largest possible diamond to the next open point within a triangular grid of side n. Each new diamond is added to the preceding structure of diamonds. At each step n, a new row of n open points is first added, extending the triangular grid.
Then the next open point is defined as the first open point encountered when the triangle is read by rows starting from the top row. a(n) is then the row position of the next open point.
Finally, starting at this open point the largest diamond that does not overlap any previous diamonds and fits within the triangular grid is added. Each diamond of side length k must cover exactly k^2 points, with the top corner on an open point. The points covered by the added diamond are then considered closed.
Is there a pattern for the values of n where a(n) = 1?

			Here zeros are the open points; closed points covered by the n-th diamond are replaced with n.
  ---------------------
  n=4       1          First a new row of 4 open points is added.
           2 3         Then the next open point is T(3,1) so a(4) = 1.
          4 0 0        Finally, the largest diamond fitting at T(3,1) is 1.
         0 0 0 0
  ---------------------
  n=5       1          First a new row of 5 open points is added.
           2 3         Then the next open point is T(3,2) so a(5) = 2.
          4 5 0        Finally, the largest diamond fitting at T(3,2) is 2.
         0 5 5 0
        0 0 5 0 0
  ---------------------
  n=6       1          First a new row of 6 open points is added.
           2 3         Then the next open point is T(3,3) so a(6) = 3.
          4 5 6        Finally, the largest diamond fitting at T(3,3) is 1.
         0 5 5 0
        0 0 5 0 0
       0 0 0 0 0 0

John Tyler Rascoe, Table of n, a(n) for n = 1..10000
John Tyler Rascoe, Python program

Cf. A045996, A186434, A194136, A239567, A279413.

Python
```
# see linked program
```

cp's OEIS Frontend

A239572 Triangle T(n, k) = Numbers of non-equivalent (mod D_3) ways to place k points on a triangular grid of side n so that no two of them are adjacent. Triangle read by rows.

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A239568 Number of ways to place 2 points on a triangular grid of side n so that they are not adjacent.

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A239569 Number of ways to place 3 points on a triangular grid of side n so that no two of them are adjacent.

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A239570 Number of ways to place 4 points on a triangular grid of side n so that no two of them are adjacent.

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A239571 Number of ways to place 5 points on a triangular grid of side n so that no two of them are adjacent.

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A282998 Number of ways to place 6 points on a triangular grid of side n so that no two of them are adjacent.

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A239438 Maximal number of points that can be placed on a triangular grid of side n so that there is no pair of adjacent points.

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