A289225 -id:A289225 - cp's OEIS frontend

A289222 Triangle read by rows: T(n, k) is the number of 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.

1, 1, 1, 1, 4, 0, 1, 9, 12, 4, 1, 16, 66, 82, 13, 0, 1, 25, 204, 670, 859, 420, 76, 0, 1, 36, 480, 3028, 9585, 15108, 10956, 2910, 231, 2, 1, 49, 960, 9780, 56520, 190371, 371016, 404746, 235380, 68793, 9030, 252, 0, 1, 64, 1722, 25574, 231635, 1336320, 4988324

Offset: 1

Heinrich Ludwig, Jul 03 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 regarded as different. If they are not to be counted, see A289229.

			The triangle begins:
  1;
  1,  1;
  1,  4,   0;
  1,  9,  12,    4;
  1, 16,  66,   82,   13,     0;
  1, 25, 204,  670,  859,   420,    76,    0;
  1, 36, 480, 3028, 9585, 15108, 10956, 2910, 231, 2;

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

Cf. A289229, A289233.

Columns 2 to 8: A000290, A289223, A289224, A289225, A289226, A289227, A289228.

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.

Original entry on oeis.org

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)

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

A289231 Number of nonequivalent 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, 4, 159, 1644, 9548, 38872, 125367, 342831, 829052, 1822785, 3714519, 7113539, 12935256, 22511616, 37728563, 61194888, 96446684, 148191316, 222597315, 327633979, 473466444, 672912717, 941968139, 1300402591, 1772439504, 2387521212, 3181168199, 4195941108, 5482512012

Offset: 4

Heinrich Ludwig, Jun 30 2017

Rotations and reflections of a selection are not counted. If they are to be counted see A289225.

			There are four nonequivalent ways to choose four 2 X 2 X 2 triangles (aaa, ..., ddd) from a 5 X 5 X 5 point grid:
      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
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 (4,-4,-1,-2,8,5,-14,-1,1,14,-5,-8,2,1,4,-4,1).

Cf. A289229, A289225, A117662, A289230, A289232.

concat(0, Vec(x^5*(4 + 143*x + 1024*x^2 + 3612*x^3 + 7423*x^4 + 10001*x^5 + 8395*x^6 + 3273*x^7 - 1362*x^8 - 2393*x^9 - 878*x^10 + 488*x^11 + 539*x^12 + 101*x^13 - 89*x^14 - 41*x^15) / ((1 - x)^9*(1 + x)^2*(1 + x + x^2)^3) + O(x^40))) \\ Colin Barker, Jun 30 2017

a(n) = (n^8 -8*n^7 -50*n^6 +556*n^5 +261*n^4 -12724*n^3 +19088*n^2 +86016*n -201024)/144 + IF(MOD(n, 2) = 1, -2*n +5)/4 + IF(MOD(n, 3) = 1, -n^2 +2*n +12)/9.

G.f.: x^5*(4 + 143*x + 1024*x^2 + 3612*x^3 + 7423*x^4 + 10001*x^5 + 8395*x^6 + 3273*x^7 - 1362*x^8 - 2393*x^9 - 878*x^10 + 488*x^11 + 539*x^12 + 101*x^13 - 89*x^14 - 41*x^15) / ((1 - x)^9*(1 + x)^2*(1 + x + x^2)^3). - Colin Barker, Jun 30 2017

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A289222 Triangle read by rows: T(n, k) is the number of 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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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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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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A289231 Number of nonequivalent 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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