A289222 -id:A289222 - cp's OEIS frontend

A289224 Number of ways to select 3 disjoint point triples from an n X n X n triangular point grid, each point triple forming a 2 X 2 X 2 triangle.

0, 4, 82, 670, 3028, 9780, 25574, 57862, 117800, 221268, 390010, 652894, 1047292, 1620580, 2431758, 3553190, 5072464, 7094372, 9743010, 13163998, 17526820, 23027284, 29890102, 38371590, 48762488, 61390900, 76625354, 94877982, 116607820, 142324228, 172590430, 208027174

Offset: 3

Heinrich Ludwig, Jun 28 2017

Rotations and reflections of a selection are regarded as different. For the number of congruence classes see A289230.

			There are four ways to choose three 2 X 2 X 2 triangles (aaa, bbb, ccc) from a 4 X 4 X 4 point grid, for example:
      a           a
     a a         a a
    b c c       b . c
   b b c .     b b c c
The other 2 possible selections are rotations of the first one.
Note: aaa, bbb, ccc are not distinguishable, they are denoted differently for a better perception of the 2 X 2 X 2 triangles only.

Heinrich Ludwig, Table of n, a(n) for n = 3..100
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A289222, A289223, A289225, A289226, A289227, A289228, A289230.

A289224:=n->(n^6-6*n^5-24*n^4+208*n^3-67*n^2-1684*n+2712)/6: seq(A289224(n), n=3..50); # Wesley Ivan Hurt, Jun 28 2017

Table[(n^6 - 6 n^5 - 24 n^4 + 208 n^3 - 67 n^2 - 1684 n + 2712)/6, {n, 3, 34}] (* or *)
LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {0, 4, 82, 670, 3028, 9780, 25574}, 32] (* or *)
Drop[#, 3] &@ CoefficientList[Series[2 x^4*(2 + 27 x + 90 x^2 - 40 x^3 - 38 x^4 + 19 x^5)/(1 - x)^7, {x, 0, 34}], x] (* Michael De Vlieger, Jun 29 2017 *)

a(n) = (n^6 - 6*n^5 - 24*n^4 + 208*n^3 - 67*n^2 - 1684*n + 2712)/6 \\ Charles R Greathouse IV, Jun 28 2017

concat(0, Vec(2*x^4*(2 + 27*x + 90*x^2 - 40*x^3 - 38*x^4 + 19*x^5) / (1 - x)^7 + O(x^60))) \\ Colin Barker, Jun 29 2017

a(n) = (n^6 -6*n^5 -24*n^4 +208*n^3 -67*n^2 -1684*n +2712)/6.
From Colin Barker, Jun 29 2017: (Start)
G.f.: 2*x^4*(2 + 27*x + 90*x^2 - 40*x^3 - 38*x^4 + 19*x^5) / (1 - x)^7.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n>9.
(End)

A289225 Number of ways to select 4 disjoint point triples from an n X n X n triangular point grid, each point triple forming a 2 X 2 X 2 triangle.

Original entry on oeis.org

0, 13, 859, 9585, 56520, 231635, 749223, 2051819, 4965960, 10924065, 22268395, 42654733, 77575104, 135020535, 226306535, 367085655, 578573168, 889013589, 1335417435, 1965599305, 2840550040, 4037177403, 5651451399, 7801992035, 10634139000, 14324544425, 19086331563

Offset: 4

Heinrich Ludwig, Jun 29 2017

Rotations and reflections of a selection are regarded as different. For the number of congruence classes see A289231.

			There are thirteen ways to choose four 2 X 2 X 2 triangles (aaa, ..., ddd) from a 5 X 5 X 5 point grid, for example:
      a           a           a           .
     a a         a a         a a         a a
    b c c       . d .       . . .       . a .
   b b c d     b d d c     b c c d     b c c d
  . . . d d   b b . c c   b b c d d   b b c d d
The other nine possible selections are rotations and reflections of these.
Note: aaa, ..., ddd are not distinguishable, they are denoted differently for a better perception of the 2 X 2 X 2 triangles only.

Heinrich Ludwig, Table of n, a(n) for n = 4..100
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A289222, A289223, A289224, A289226, A289227, A289228, A289231.

A289225:=n->(n^8 -8*n^7 -50*n^6 +556*n^5 +231*n^4 -12388*n^3 +17914*n^2 +86648*n -198528)/24: seq(A289225(n), n=4..50); # Wesley Ivan Hurt, Jun 29 2017

concat(0, Vec(x^5*(13 + 742*x + 2322*x^2 + 87*x^3 - 2503*x^4 + 684*x^5 + 560*x^6 - 225*x^7) / (1 - x)^9 + O(x^30))) \\ Colin Barker, Jun 30 2017

a(n) = (n^8 -8*n^7 -50*n^6 +556*n^5 +231*n^4 -12388*n^3 +17914*n^2 +86648*n -198528)/24.
From Colin Barker, Jun 30 2017: (Start)
G.f.: x^5*(13 + 742*x + 2322*x^2 + 87*x^3 - 2503*x^4 + 684*x^5 + 560*x^6 - 225*x^7) / (1 - x)^9.
a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9) for n>12.
(End)

A289226 Number of ways to select 5 disjoint point triples from an n X n X n triangular point grid, each point triple forming a 2 X 2 X 2 triangle.

Original entry on oeis.org

0, 420, 15108, 190371, 1336320, 6528948, 24951780, 79851975, 223419840, 562591836, 1301255556, 2806131075, 5705746752, 11034449244, 20436317412, 36447218199, 62877079680, 105318792564, 171815016708, 273719593923, 426796282752, 652604165220, 980226360036, 1448406641607

Offset: 5

Heinrich Ludwig, Jul 01 2017

Rotations and reflections of a selection are regarded as different. For the number of congruence classes see A289232.

			There are 420 ways to choose five 2 X 2 X 2 triangles (aaa, ..., eee) from a 6 X 6 X 6 point grid, for example:
        .               a
       . .             a a
      . . .           . d .
     a a b b         b d d c
    c a d b e       b b e c c
   c c d d e e     . . e e . .
Note: aaa, ..., eee are not distinguishable, they are denoted differently for a better perception of the 2 X 2 X 2 triangles only.

Heinrich Ludwig, Table of n, a(n) for n = 5..100
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A289222, A289223, A289224, A289225, A289227, A289228, A289232.

concat(0, Vec(3*x^6*(140 + 3496*x + 15761*x^2 + 1293*x^3 - 18129*x^4 + 3779*x^5 + 6103*x^6 - 1637*x^7 - 1139*x^8 + 413*x^9) / (1 - x)^11 + O(x^40))) \\ Colin Barker, Jul 01 2017

a(n) = (n^10 -10*n^9 -85*n^8 +1160*n^7 +1345*n^6 -49162*n^5 +62145*n^4 +892140*n^3 -2198566*n^2 -5725008*n +18190440)/120.

G.f.: 3*x^6*(140 + 3496*x + 15761*x^2 + 1293*x^3 - 18129*x^4 + 3779*x^5 + 6103*x^6 - 1637*x^7 - 1139*x^8 + 413*x^9) / (1 - x)^11. - Colin Barker, Jul 01 2017

A289223 Number of ways to select 2 disjoint point triples from an n X n X n triangular point grid, each point triple forming an 2 X 2 X 2 triangle.

Original entry on oeis.org

0, 0, 12, 66, 204, 480, 960, 1722, 2856, 4464, 6660, 9570, 13332, 18096, 24024, 31290, 40080, 50592, 63036, 77634, 94620, 114240, 136752, 162426, 191544, 224400, 261300, 302562, 348516, 399504, 455880, 518010, 586272, 661056, 742764, 831810, 928620, 1033632, 1147296

Offset: 2

Heinrich Ludwig, Jun 28 2017

Rotations and reflections of a selection are regarded as different. For the number of congruence classes see A117662(n-1).

			There are 12 ways to choose two 2 X 2 X 2 triangles (xxx) from a 4 X 4 X 4 point grid, for example:
      x           x          x
     x x         x x        x x
    . x x       x . .      . x .
   . . x .     x x . .    . x x .
The other nine selections are reflections or rotations of these three.

Heinrich Ludwig, Table of n, a(n) for n = 2..100
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A117662, A289222, A289224, A289225, A289226, A289227, A289228.

Vec(6*x^4*(2 - x)*(1 + x) / (1 - x)^5 + O(x^60)) \\ Colin Barker, Jun 28 2017

a(n) = (n^4 -4*n^3 -7*n^2 +46*n -48)/2 for n>=2.
From Colin Barker, Jun 28 2017: (Start)
G.f.: 6*x^4*(2 - x)*(1 + x) / (1 - x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>6.
(End)

A289227 Number of ways to select 6 disjoint point triples from an n X n X n triangular point grid, each point triple forming a 2 X 2 X 2 triangle.

Original entry on oeis.org

0, 76, 10956, 371016, 4988324, 39302784, 218633416, 952344088, 3460482612, 10932805668, 30901640212, 79762409256, 190898410020, 428596770008, 910935932112, 1846146025240, 3588666200596, 6723331905852, 12188915557404, 21455723224456, 36776237135268, 61533021405936

Offset: 5

Heinrich Ludwig, Jul 01 2017

Rotations and reflections of a selection are regarded as different.

			There are 76 ways to choose six 2 X 2 X 2 triangles (aaa, ..., fff) from a 6 X 6 X 6 point grid, for example:
        a               a
       a a             a a
      . . .           b . c
     b b c c         b b c c
    d b e c f       d . e . f
   d d e e f f     d d e e f f
Note: aaa, ..., fff are not distinguishable, they are denoted differently for a better perception of the 2 X 2 X 2 triangles only.

Heinrich Ludwig, Table of n, a(n) for n = 5..100
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Cf. A289222, A289223, A289224, A289225, A289226, A289228.

concat(0, Vec(4*x^6*(19 + 2492*x + 58629*x^2 + 249487*x^3 + 78686*x^4 - 397088*x^5 + 93163*x^6 + 160960*x^7 - 77014*x^8 - 10728*x^9 + 4312*x^10 + 5013*x^11 - 1611*x^12) / (1 - x)^13 + O(x^40))) \\ Colin Barker, Jul 01 2017

a(n) = (n^12 -12*n^11 -129*n^10 +2090*n^9 +3985*n^8 -142832*n^7 +152809*n^6 +4752598*n^5 -12392266*n^4 -76011076*n^3 +274393360*n^2 +455879232*n -2015187840)/720 for n>=6.

G.f.: 4*x^6*(19 + 2492*x + 58629*x^2 + 249487*x^3 + 78686*x^4 - 397088*x^5 + 93163*x^6 + 160960*x^7 - 77014*x^8 - 10728*x^9 + 4312*x^10 + 5013*x^11 - 1611*x^12) / (1 - x)^13. - Colin Barker, Jul 01 2017

A289228 Number of ways to select 7 disjoint point triples from an n X n X n triangular point grid, each point triple forming a 2 X 2 X 2 triangle.

Original entry on oeis.org

0, 2910, 404746, 12025068, 165279612, 1405048082, 8605979390, 41555851716, 167529980320, 586136559350, 1829074790082, 5193890370940, 13625393372916, 33410188057962, 77284672892438, 169909353488372, 357177283295160, 721559475338446, 1406717921047994, 2656028041092876

Offset: 6

Heinrich Ludwig, Jul 01 2017

Rotations and reflections of a selection are regarded as different.

			There are 2910 ways to choose seven 2 X 2 X 2 triangles (aaa, ..., ggg) from a 7 X 7 X 7 point grid, for example:
          a
         a a
        b . c
       b b c c
      d d . e e
     f d . . e g
    f f . . . g g
Note: aaa, ..., ggg are not distinguishable, they are denoted differently for a better perception of the 2 X 2 X 2 triangles only.

Heinrich Ludwig, Table of n, a(n) for n = 6..100
Index entries for linear recurrences with constant coefficients, signature (15,-105,455,-1365,3003,-5005,6435,-6435,5005,-3003,1365,-455,105,-15,1).

Cf. A289222, A289223, A289224, A289225, A289226, A289227.

concat(0, Vec(2*x^7*(1455 + 180548*x + 3129714*x^2 + 13038936*x^3 + 4149381*x^4 - 21524480*x^5 + 3658074*x^6 + 12791138*x^7 - 6864973*x^8 - 1299402*x^9 + 1667400*x^10 - 272962*x^11 + 37953*x^12 - 60178*x^13 + 16036*x^14) / (1 - x)^15 + O(x^40))) \\ Colin Barker, Jul 01 2017

a(n) = (n^14 -14*n^13 -182*n^12 +3416*n^11 +9072*n^10 -342062*n^9 +296688*n^8 +17893944*n^7 -48845153*n^6 -511039228*n^5 +2041220174*n^4 +7429535400*n^3 -37737333320*n^2 -41483946096*n +262680697440)/5040 for n>=7.

G.f.: 2*x^7*(1455 + 180548*x + 3129714*x^2 + 13038936*x^3 + 4149381*x^4 - 21524480*x^5 + 3658074*x^6 + 12791138*x^7 - 6864973*x^8 - 1299402*x^9 + 1667400*x^10 - 272962*x^11 + 37953*x^12 - 60178*x^13 + 16036*x^14) / (1 - x)^15. - Colin Barker, Jul 01 2017

A289229 Triangle read by rows: T(n, k) is the number of nonequivalent ways to select k disjoint point triples from an n X n X n triangular point grid, each point triple forming a 2 X 2 X 2 triangle.

Original entry on oeis.org

1, 1, 1, 1, 2, 0, 1, 3, 3, 2, 1, 5, 14, 19, 4, 0, 1, 7, 40, 127, 159, 77, 17, 0, 1, 9, 90, 536, 1644, 2569, 1876, 500, 42, 1, 1, 12, 175, 1688, 9548, 31951, 62171, 67765, 39459, 11579, 1547, 47, 0, 1, 15, 308, 4357, 38872, 223346, 832628, 2005948, 3072004, 2897626

Offset: 1

Heinrich Ludwig, Jul 04 2017

The row index starts from 1. The column index k runs from 0 to floor(n*(n+1)/6), which is a trivial upper bound for the maximal number of 2 X 2 X 2 triangles that can be selected from an n X n X n triangular grid.

Rotations and reflections of a selection are not counted. If they are to be counted, see A289222.

			The triangle begins:
  1;
  1,  1;
  1,  2,   0;
  1,  3,   3,    3;
  1,  5,  14,   19,    4,     0;
  1,  7,  40,  127,  159,    77,    17,     0;
  1,  9,  90,  536, 1644,  2569,  1876,   500,    42,     1;
  1, 12, 175, 1688, 9548, 31951, 62171, 67765, 39459, 11579, 1547, 47, 0;

Heinrich Ludwig, Table of n, a(n) for n = 1..116, first 11 (and a half) rows of the triangular array

Cf. A289222, A289233.

Columns 2 to 6: A001840, A117662, A289230, A289231, A289232.

A289233 Largest number of disjoint point triples that can be chosen from an n X n X n triangular point grid, each point triple forming a 2 X 2 X 2 triangle.

Original entry on oeis.org

0, 1, 1, 3, 4, 6, 9, 11, 15, 18, 22, 26, 30, 35, 39, 45, 50, 56, 63, 69, 77, 84, 92, 100, 108, 117, 125, 135, 144, 154, 165, 175, 187, 198, 210, 222, 234, 247, 259, 273, 286, 300, 315, 329, 345, 360, 376, 392, 408, 425, 441, 459, 476, 494, 513, 531, 551, 570

Offset: 1

Heinrich Ludwig, Jul 08 2017

nonn

A001840(n-1) = floor(n*(n+1)/6) is a trivial upper bound for a(n), which is not reached for n = 3, 5, 6, 8, 15, 17, 18, 20 but for all other n <= 21.
From Jon E. Schoenfield, Aug 16 2017: (Start)
If n is even, then the bottom three rows of points can be assembled into n-1 of the 3-point triangles; e.g., the full grid for n = 8 has 8*9/2 = 36 points, of which the 21 points in the bottom three rows can be assembled into 7 three-point triangles as follows, leaving the remaining 15 points in the same triangular arrangement as in the full grid for the n = 5 case:
.
Row 1:                .
.
Row 2:              .   .
.
Row 3:            .   .   .
.
Row 4:          .   .   .   .
.
Row 5:        .   .   .   .   .
.
Row 6:      2---2   4---4   6---6
.            \ /     \ /     \ /
Row 7:    1   2   3   4   5   6   7
.        / \     / \     / \     / \
Row 8:  1---1   3---3   5---5   7---7
.
Thus, if n is even, then a(n) >= a(n-3) + n - 1.
Similarly, if n mod 3 = 2, then the bottom two rows of points can be assembled into (2n-1)/3 of the 3-point triangles; e.g., of the 36 points in the full grid for n = 8, the 15 points in the bottom two rows can be assembled into 5 three-point triangles as follows, leaving the remaining 21 points in the same triangular arrangement as in the full grid for the n=6 case:
.
Row 1:                     .
.
Row 2:                   .   .
.
Row 3:                 .   .   .
.
Row 4:               .   .   .   .
.
Row 5:             .   .   .   .   .
.
Row 6:           .   .   .   .   .   .
.
Row 7:         1   2---2   3   4---4   5
.             / \   \ /   / \   \ /   / \
Row 8:       1---1   2   3---3   4   5---5
.
Thus, if n mod 3 = 2, then a(n) >= a(n-2) + (2n-1)/3.
Given the terms through a(21) = 77, the two lower bounds above and the upper bound a(n) <= floor(n(n+1)/6) are sufficient to establish that a(22) = 84, a(23) = 92, a(24) = 100, a(26) = 117, and a(28) = 135. (A solution with 108 three-point triangles can be constructed on the n = 25 grid.)
Conjecture: a(n) = floor((n(n+1) - floor(((n+3) mod 12)/6))/6); i.e., the upper bound floor(n(n+1)/6) can be reached unless n(n+1)/2 is a multiple of 3 and (n+3) mod 12 >= 6, in which case a(n) falls short of the upper bound by 1. (End)
a(31) = 165, a(33) = 187, a(34) = 198, a(35) = 210, a(36) = 222, a(37) = 234, a(38) = 247, and a(40) = 273. - Rob Pratt, Dec 19 2017
From Jon E. Schoenfield, Dec 21 2017: (Start)
If we refer to a triangular grid with n points on each side simply as an "n-triangle", then for any n > 13, the n-triangle can be broken into an (n-12)-triangle, a 12-triangle, and a parallelogram-shaped grid with 12 points on each of two opposite sides and n-12 points on each of the other two sides (with n-12 > 1). E.g., we can break the 21-triangle into (1) a 9-triangle, (2) a 12-triangle, and (3) a 9 X 12 parallelogram grid:
                         _________
                        1          ^
                       1 1         |
                      1 1 1        |
                     1 1 1 1       |
                    1 1 1 1 1      9
                   1 1 1 1 1 1     |
                  1 1 1 1 1 1 1    |
                 1 1 1 1 1 1 1 1   |
                1 1 1 1 1 1 1 1 1__v
                                  _______________
              2  3 3 3 3 3 3 3 3 3                ^
             2 2  3 3 3 3 3 3 3 3 3               |
            2 2 2  3 3 3 3 3 3 3 3 3              |
           2 2 2 2  3 3 3 3 3 3 3 3 3             |
          2 2 2 2 2  3 3 3 3 3 3 3 3 3            |
         2 2 2 2 2 2  3 3 3 3 3 3 3 3 3          12
        2 2 2 2 2 2 2  3 3 3 3 3 3 3 3 3          |
       2 2 2 2 2 2 2 2  3 3 3 3 3 3 3 3 3         |
      2 2 2 2 2 2 2 2 2  3 3 3 3 3 3 3 3 3        |
     2 2 2 2 2 2 2 2 2 2  3 3 3 3 3 3 3 3 3       |
    2 2 2 2 2 2 2 2 2 2 2  3 3 3 3 3 3 3 3 3      |
   2 2 2 2 2 2 2 2 2 2 2 2  3 3 3 3 3 3 3 3 3 ____v
  |                       ||                 |
  |<---------12---------->||<-------9------->|
.
All points of a 2 X 12 parallelogram grid and all points of a 3 X 12 parallelogram grid can be assembled into 3-point triangles using the simple patterns
.
         o o                     o o .
          o .                     o . .
           . .                     o o .
            o o                     o . .
             o .                     o o .
              . .         and         o . .
               o o                     o o .
                o .                     o . .
                 . .                     o o .
                  o o                     o . .
                   o .                     o o .
                    . .                     o . .
.
respectively, so those patterns can be combined to assemble a k X 12 parallelogram completely into 3-point triangles for any k > 1. Thus, since both the 12-triangle and the (n-12) X 12 parallelogram grid can be completely assembled into 3-point triangles for any n > 13, we have, for n > 13, a(n) >= a(n-12) + a(12) + (n-12)*12/3, which reduces to a(n) >= a(n-12) + 4n - 22. In particular, if the trivial upper bound floor(n*(n+1)/6) is reached by a(n), then it is also reached by a(n+12k) for any positive integer k. Given a(1)..a(12), this is sufficient to prove that a(n) = floor(n*(n+1)/6) except when n mod 12 is 3, 5, 6, or 8; in those cases, a(n) >= floor(n*(n+1)/6) - 1. (End)
Theorem 1.1 of the Conway and Lagarias link indicates that all the points can be covered by 3-point triangles iff n mod 12 = 0, 2, 9, or 11. That fact, together with the above results for a(n) in the specific cases for n in [1, 3..8, 10] and the recursive lower bound a(n) >= a(n-12) + 4n - 22, is sufficient to prove that a(n) = floor(n*(n+1)/6) - d where d is 1 when n mod 12 is in [3, 5, 6, 8] and 0 otherwise. - Jon E. Schoenfield, Dec 26 2017

			From a 21 X 21 X 21 point grid up to 77 disjoint 2 X 2 X 2 triangles (aaa, bbb, ...) can be chosen. Selections like the one below with no point left are very rare compared to C(400, 77). 400 is the total number of 2 X 2 X 2 triangles in the 21-grid.
                        a
                       a a
                      c c b
                     d c b b
                    d d f f e
                   h h g f e e
                  i h g g k k j
                 i i l m m k j j
                p p l l m n q q o
               r p s t t n n q o o
              r r s s t u w w x x v
             y a a b b u u w z x v v
            y y a c b d f f z z g g e
           j j h c c d d f i k k g e e
          l j h h m m n n i i k o o p p
         l l w w q m r n s x x t o u p v
        y z z w q q r r s s x t t u u v v
       y y z f f a g g b h h c i i d j j e
      k k l l f a a g b b h c c i d d j e e
     m k n l o u u p v v q w w r x x s y y t
    m m n n o o u p p v q q w r r x s s y t t

J. H. Conway and J. C. Lagarias, Tiling with Polyominoes and Combinatorial Group Theory, JCTA 53 (1990), 183-208.
Index entries for linear recurrences with constant coefficients, signature (2, 0, -1, -2, 2, 1, 0, -2, 1).

Cf. A001840, A289222, A289229.

f[n_] := Floor[n (n +1)/6] - If[ !MemberQ[{3, 5, 6, 8}, Mod[n, 12]], 0, 1]; Array[f, 58] (* or *)
CoefficientList[ Series[(-x +x^2 -x^3 +x^4 -x^5)/((-1 +x)^3 (1 +x -x^3 +x^5 +x^6)), {x, 0, 57}], x] (* or *)
LinearRecurrence[{2, 0, -1, -2, 2, 1, 0, -2, 1}, {0, 1, 1, 3, 4, 6, 9, 11, 15}, 58] (* Robert G. Wilson v, Dec 26 2017 *)

G.f.: x*(1 - x + x^2 - x^3 + x^4) / ((1 - x)^3*(1 + x + x^2)*(1 - x^2 + x^4)) (conjectured). - Colin Barker, Jul 08 2017

a(n) = floor(n*(n+1)/6) except when n mod 12 is 3, 5, 6, or 8; in those cases, a(n) = floor(n*(n+1)/6) - 1. - Jon E. Schoenfield, Dec 25 2017

a(22)-a(26) from Jon E. Schoenfield, Aug 16 2017
a(27)-a(28) from Rob Pratt, Dec 19 2017
More terms from Jon E. Schoenfield, Dec 25 2017

cp's OEIS Frontend

A289224 Number of ways to select 3 disjoint point triples from an n X n X n triangular point grid, each point triple forming a 2 X 2 X 2 triangle.

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A289225 Number of ways to select 4 disjoint point triples from an n X n X n triangular point grid, each point triple forming a 2 X 2 X 2 triangle.

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A289226 Number of ways to select 5 disjoint point triples from an n X n X n triangular point grid, each point triple forming a 2 X 2 X 2 triangle.

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A289223 Number of ways to select 2 disjoint point triples from an n X n X n triangular point grid, each point triple forming an 2 X 2 X 2 triangle.

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A289227 Number of ways to select 6 disjoint point triples from an n X n X n triangular point grid, each point triple forming a 2 X 2 X 2 triangle.

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A289228 Number of ways to select 7 disjoint point triples from an n X n X n triangular point grid, each point triple forming a 2 X 2 X 2 triangle.

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A289229 Triangle read by rows: T(n, k) is the number of nonequivalent ways to select k disjoint point triples from an n X n X n triangular point grid, each point triple forming a 2 X 2 X 2 triangle.

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A289233 Largest number of disjoint point triples that can be chosen from an n X n X n triangular point grid, each point triple forming a 2 X 2 X 2 triangle.