A289231 -id:A289231 - cp's OEIS frontend

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.

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)

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.

A289230 Number of nonequivalent 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, 2, 19, 127, 536, 1688, 4357, 9789, 19844, 37172, 65397, 109335, 175214, 270934, 406329, 593463, 846934, 1184212, 1625979, 2196509, 2924050, 3841240, 4985531, 6399647, 8132044, 10237410, 12777167, 15820007, 19442436, 23729352, 28774625, 34681717, 41564304, 49546932

Offset: 3

Heinrich Ludwig, Jun 30 2017

Rotations and reflections of a selection are not counted. If they are to be counted see A289224.

			There are two nonequivalent ways to choose three 2 X 2 X 2 triangles (aaa, bbb, ccc) from a 4 X 4 X 4 point grid:
      a           a
     a a         a a
    b c c       b . c
   b b c .     b b c c
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 (5,-9,6,0,0,0,-6,9,-5,1).

Cf. A289224, A289229, A117662, A289231, A289232.

concat(0, Vec(x^4*(2 + 9*x + 50*x^2 + 60*x^3 + 37*x^4 - 21*x^5 - 20*x^6 - 4*x^7 + 7*x^8) / ((1 - x)^7*(1 + x)*(1 + x + x^2)) + O(x^40))) \\ Colin Barker, Jun 30 2017

a(n) = (n^6 -6*n^5 -24*n^4 +220*n^3 -153*n^2 -1488*n +2592)/36 + IF(MOD(n, 2) = 1, -1)/2 + IF(MOD(n, 3) = 1, -2)/9.

G.f.: x^4*(2 + 9*x + 50*x^2 + 60*x^3 + 37*x^4 - 21*x^5 - 20*x^6 - 4*x^7 + 7*x^8) / ((1 - x)^7*(1 + x)*(1 + x + x^2)). - Colin Barker, Jun 30 2017

A289232 Number of nonequivalent 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, 77, 2569, 31951, 223346, 1089665, 4161705, 13314461, 37246668, 93781829, 216901737, 467727523, 951014654, 1839155785, 3406165049, 6074688977, 10479716856, 17553399741, 28636182537, 45620375447, 71133273514, 108768061009, 163371926729, 241402171109, 351362501892

Offset: 5

Heinrich Ludwig, Jul 01 2017

Rotations and reflections of a selection are not counted. If they are to be counted see A289226.

			There are 77 nonequivalent 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 (8,-25,32,11,-88,99,0,-99,88,-11,-32,25,-8,1).

Cf. A289229, A117662, A289230, A289231, A289226.

concat(0, Vec(x^6*(77 + 1953*x + 13324*x^2 + 29499*x^3 + 18617*x^4 - 15880*x^5 - 17638*x^6 + 4876*x^7 + 8057*x^8 - 881*x^9 - 1966*x^10 + 81*x^11 + 201*x^12) / ((1 - x)^11*(1 + x)^3) + O(x^40))) \\ Colin Barker, Jul 01 2017

a(n) = (n^10 -10*n^9 -85*n^8 +1160*n^7 +1345*n^6 -49084*n^5 +61035*n^4 +897210*n^3 -2205196*n^2 -5725656*n +18174960)/720 + IF(MOD(n, 2) = 1, -2*n^2 +13*n -11)/4.

G.f.: x^6*(77 + 1953*x + 13324*x^2 + 29499*x^3 + 18617*x^4 - 15880*x^5 - 17638*x^6 + 4876*x^7 + 8057*x^8 - 881*x^9 - 1966*x^10 + 81*x^11 + 201*x^12) / ((1 - x)^11*(1 + x)^3). - Colin Barker, Jul 01 2017

cp's OEIS Frontend

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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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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A289230 Number of nonequivalent 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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A289232 Number of nonequivalent 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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