A239574 -id:A239574 - 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.

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

A239573 Number of non-equivalent (mod D_3) ways to place 3 indistinguishable points on a triangular grid of side n so that no two of them are adjacent.

Original entry on oeis.org

0, 1, 6, 32, 113, 329, 790, 1702, 3320, 6057, 10400, 17074, 26903, 41047, 60796, 87886, 124220, 172275, 234732, 314992, 416703, 544391, 702878, 898040, 1136098, 1424521, 1771178, 2185392, 2676947, 3257305, 3938450, 4734286, 5659306, 6730177, 7964228, 9381234

Offset: 2

Heinrich Ludwig, Mar 23 2014

Rotations and reflections of placements are not counted. If they are to be counted see A239569.

			There are a(4) = 6 non-equivalent ways to place 3 points on a triangular grid of side 4:
    .           X           X           X           X           X
   . X         . .         . .         . .         . .         . .
  X . .       X . X       X . .       X . .       . X .       . . .
. . X .     . . . .     . . X .     . . . X     . . . X     X . . X

Heinrich Ludwig, Table of n, a(n) for n = 2..1000
Index entries for linear recurrences with constant coefficients, signature (3,0,-7,3,6,0,-6,-3,7,0,-3,1)

Cf. A239572, A239569, A032091 (2 points), A239574 (4 points), A239575 (5 points), A279446 (6 points).

concat(0, Vec(-x^3*(2*x^9 +x^8 -8*x^7 -9*x^6 +3*x^5 +29*x^4 +24*x^3 +14*x^2 +3*x +1)/((x -1)^7*(x +1)^3*(x^2 +x +1)) + O(x^100))) \\ Colin Barker, Mar 23 2014

a(n) = (n^6 + 3*n^5 - 39*n^4 + 10*n^3 + 456*n^2 - 1008*n + 576)/288 + IF(MOD(n, 2) = 1)*(3*n^2 - 5*n - 5)/32 + IF(MOD(n, 3) = 1)*2/9.

G.f.: -x^3*(2*x^9 +x^8 -8*x^7 -9*x^6 +3*x^5 +29*x^4 +24*x^3 +14*x^2 +3*x +1) / ((x -1)^7*(x +1)^3*(x^2 +x +1)). - Colin Barker, Mar 23 2014

A239575 Number of non-equivalent (mod D_3) ways to place 5 indistinguishable points on a triangular grid of side n so that no two of them are adjacent.

Original entry on oeis.org

0, 0, 7, 176, 1976, 12565, 57275, 207018, 634166, 1711262, 4181915, 9428657, 19892816, 39684027, 75473209, 137721045, 242391212, 413215132, 684733527, 1106194950, 1746637600, 2701244609, 4099429895, 6114748948, 8977257362, 12988406970, 18539308619, 26132434991

Offset: 3

Heinrich Ludwig, Mar 23 2014

Rotations and reflections of placements are not counted. If they are to be counted see A239571.

			There are a(5) = 7 non-equivalent ways to place 5 points (x) on a triangular grid of side 5. These are:
        x             x             .             x
       . .           . .           . .           . .
      x . x         x . x         x . x         . x .
     . . . .       . . . .       . . . .       . . . .
    x . . . x     . x . x .     x . x . x     x . x . x
.
        x             x             x
       . .           . .           . .
      . x .         . x .         x . x
     x . . x       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 (6,-10,-10,50,-34,-66,110,0,-110,66,34,-50,10,10,-6,1)

Cf. A239572, A239571, A032091 (2 points), A239573 (3 points), A239574 (4 points), 279446 (6 points).

Table[(n^10 + 5*n^9 - 130*n^8 - 310*n^7 + 7465*n^6 - 1336*n^5 - 202980*n^4 + 464160*n^3 + 1783424*n^2 - 8360064*n + 9192960)/23040 + (1-(-1)^n)/2*(25*n^4 - 94*n^3 - 418*n^2 + 2053*n - 1779)/1536,{n,3,20}] (* Vaclav Kotesovec after Heinrich Ludwig, Mar 31 2014 *)
Drop[CoefficientList[Series[x^2*(-19 - (19 - 114*x + 190*x^2 + 197*x^3 - 816*x^4 + 1636*x^5 + 3793*x^6 + 965*x^7 + 216*x^8 + 194*x^9 - 2278*x^10 + 53*x^11 + 1547*x^12 - 336*x^13 - 351*x^14 + 125*x^15) / ((-1+x)^11*(1+x)^5)), {x, 0, 20}], x], 3] (* Vaclav Kotesovec, Mar 31 2014 *)

a(n) = (n^10 + 5*n^9 - 130*n^8 - 310*n^7 + 7465*n^6 - 1336*n^5 - 202980*n^4 + 464160*n^3 + 1783424*n^2 - 8360064*n + 9192960)/23040 + IF(MOD(n,2) = 1)*(25*n^4 - 94*n^3 - 418*n^2 + 2053*n - 1779)/1536.

G.f.: x^2*(-19 - (19 - 114*x + 190*x^2 + 197*x^3 - 816*x^4 + 1636*x^5 + 3793*x^6 + 965*x^7 + 216*x^8 + 194*x^9 - 2278*x^10 + 53*x^11 + 1547*x^12 - 336*x^13 - 351*x^14 + 125*x^15) / ((-1+x)^11 * (1+x)^5)). - Vaclav Kotesovec, Mar 31 2014

A279446 Number of non-equivalent (mod D_3) ways to place 6 indistinguishable points on a triangular grid of side n so that no two of them are adjacent.

Original entry on oeis.org

0, 0, 1, 66, 2096, 25676, 187984, 983172, 4073312, 14196011, 43309138, 118818916, 298926225, 699619679, 1540212590, 3217045155, 6419240369, 12304959047, 22763742133, 40797668697, 71065355815, 120643462032, 200077436639, 324808463585, 517088445952, 808515893580

Offset: 3

Heinrich Ludwig, Feb 26 2017

Rotations and reflections of placements are not counted. For numbers if they are to be counted see A282998.

			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 (4,0,-17,8,36,-7,-68,-18,113,52,-126,-92,92,126,-52,-113,18,68,7,-36,-8,17,0,-4,1).

Cf. A282998, A239572, A032091 (2 points), A239573 (3 points), A239574 (4 points), A239575 (5 points).

Table[Boole[n > 4] ((n^12 + 6 n^11 - 195 n^10 - 670 n^9 + 17455 n^8 + 13426 n^7 - 835256 n^6 + 1246240 n^5 + 19563664 n^4 - 68181792 n^3 - 131524224 n^2 + 969500160 n - 1298903040)/276480 + Boole[OddQ@ n] (162 n^5 - 715 n^4 - 4480 n^3 + 21955 n^2 + 1108 n - 41685)/30720 + Boole[Mod[n, 3] == 1] (n^2 + n - 25)/27), {n, 3, 28}] (* Michael De Vlieger, Feb 26 2017 *)

concat(vector(2), Vec(x^5*(1 + 62*x + 1832*x^2 + 17309*x^3 + 86394*x^4 + 266304*x^5 + 557979*x^6 + 818157*x^7 + 829988*x^8 + 519203*x^9 + 94134*x^10 - 150065*x^11 - 123434*x^12 + 7445*x^13 + 64052*x^14 + 29943*x^15 - 11247*x^16 - 15803*x^17 - 3012*x^18 + 3100*x^19 + 1722*x^20 - 15*x^21 - 233*x^22 - 56*x^23) / ((1 - x)^13*(1 + x)^6*(1 + x + x^2)^3) + 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 - 835256*n^6 + 1246240*n^5 + 19563664*n^4 - 68181792*n^3 - 131524224*n^2 + 969500160*n - 1298903040)/276480 + IF(MOD(n, 2) = 1, 162*n^5 - 715*n^4 - 4480*n^3 + 21955*n^2 + 1108*n - 41685)/30720 + IF(MOD(n, 3) = 1, n^2 + n - 25)/27 for n>=4.

G.f.: x^5*(1 + 62*x + 1832*x^2 + 17309*x^3 + 86394*x^4 + 266304*x^5 + 557979*x^6 + 818157*x^7 + 829988*x^8 + 519203*x^9 + 94134*x^10 - 150065*x^11 - 123434*x^12 + 7445*x^13 + 64052*x^14 + 29943*x^15 - 11247*x^16 - 15803*x^17 - 3012*x^18 + 3100*x^19 + 1722*x^20 - 15*x^21 - 233*x^22 - 56*x^23) / ((1 - x)^13*(1 + x)^6*(1 + x + x^2)^3). - Colin Barker, Feb 26 2017

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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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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A239573 Number of non-equivalent (mod D_3) ways to place 3 indistinguishable points on a triangular grid of side n so that no two of them are adjacent.

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A239575 Number of non-equivalent (mod D_3) ways to place 5 indistinguishable points on a triangular grid of side n so that no two of them are adjacent.

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A279446 Number of non-equivalent (mod D_3) ways to place 6 indistinguishable points on a triangular grid of side n so that no two of them are adjacent.

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