A279446 -id:A279446 - 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.

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

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

A239574 Number of non-equivalent (mod D_3) ways to place 4 indistinguishable points on a triangular grid of side n so that no two of them are adjacent.

Original entry on oeis.org

0, 1, 24, 200, 1053, 3932, 11988, 31298, 73046, 155880, 310046, 581414, 1038634, 1779531, 2942114, 4714412, 7350595, 11184786, 16654116, 24317554, 34886940, 49252544, 68523846, 94062350, 127534794, 170954603, 226748678, 297809946, 387580007, 500113190, 640178710

Offset: 3

Heinrich Ludwig, Mar 23 2014

Rotations and reflections of placements are not counted. If they are to be counted see A239570.

			There is a(4) = 1 way to place 4 points on a triangular grid of side n = 4:
      X
     . .
    . X .
   X . . X

Heinrich Ludwig, Table of n, a(n) for n = 3..1000
Index entries for linear recurrences with constant coefficients, signature (2,3,-5,-8,3,19,4,-24,-15,15,24,-4,-19,-3,8,5,-3,-2,1)

Cf. A239572, A239570, A032091 (2 points), A239573 (3 points), A239575 (5 points), A279446 (6 points).

Drop[CoefficientList[Series[x^4*(-1 - 22*x - 149*x^2 - 586*x^3 - 1354*x^4 - 2154*x^5 - 2300*x^6 - 1510*x^7 - 259*x^8 + 470*x^9 + 443*x^10 + 70*x^11 - 130*x^12 - 94*x^13 - 10*x^14 + 18*x^15 + 8*x^16) / ((-1+x)^9 * (1+x)^4 * (1+x+x^2)^3), {x, 0, 20}], x],3] (* Vaclav Kotesovec, Mar 29 2014 *)
Table[(n^8+4*n^7-78*n^6-104*n^5+2556*n^4-3152*n^3-27280*n^2+89664*n-78336)/2304 + If[Mod[n,2]==1,(28*n^3-54*n^2-160*n+129)/768,0] + If[Mod[n,3]==1,(n^2+n-14)/18,0],{n,3,20}] (* Vaclav Kotesovec after Heinrich Ludwig, Mar 29 2014 *)

a(n) = (n^8 +4*n^7 -78*n^6 -104*n^5 +2556*n^4 -3152*n^3 -27280*n^2 +89664*n -78336)/2304 +IF(n == 1 mod 2)*(28*n^3 -54*n^2 -160*n +129)/768 +IF(n == 1 mod 3)*(n^2 +n -14)/18.

G.f.: x^4*(-1 - 22*x - 149*x^2 - 586*x^3 - 1354*x^4 - 2154*x^5 - 2300*x^6 - 1510*x^7 - 259*x^8 + 470*x^9 + 443*x^10 + 70*x^11 - 130*x^12 - 94*x^13 - 10*x^14 + 18*x^15 + 8*x^16) / ((-1+x)^9 * (1+x)^4 * (1+x+x^2)^3). - Vaclav Kotesovec, Mar 29 2014

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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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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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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A239574 Number of non-equivalent (mod D_3) ways to place 4 indistinguishable points on a triangular grid of side n so that no two of them are adjacent.

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