cp's OEIS Frontend

A239567 Triangle T(n, k) = Numbers of ways to place k points on a triangular grid of side n so that no two of them are adjacent. Triangle read by rows.

%I A239567 #36 Feb 16 2025 08:33:21
%S A239567 1,3,6,6,1,10,27,21,1,15,75,151,114,27,1,21,165,615,1137,999,353,27,
%T A239567 28,315,1845,6100,11565,12231,6715,1686,150,2,36,546,4571,23265,74811,
%U A239567 153194,196899,153072,67229,14727,1257,28,45,882,9926,71211,342042,1124820
%N A239567 Triangle T(n, k) = Numbers of ways to place k points on a triangular grid of side n so that no two of them are adjacent. Triangle read by rows.
%C A239567 The triangle T(n, k) is irregularly shaped: 1 <= k <= A239438(n). First row corresponds to n = 1.
%C A239567 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).
%C A239567 Row n is the coefficients of the independence polynomial of the triangular grid graph, omitting x^0 coefficients. - _Eric W. Weisstein_, Nov 11 2016
%H A239567 Heinrich Ludwig, <a href="/A239567/b239567.txt">Table of n, a(n) for n = 1..136</a>
%H A239567 Stan Wagon, <a href="http://www.jstor.org/stable/10.4169/college.math.j.45.4.278">Graph Theory Problems from Hexagonal and Traditional Chess</a>, The College Mathematics Journal, Vol. 45, No. 4, September 2014, pp. 278-287
%H A239567 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/IndependencePolynomial.html">Independence Polynomial</a>
%H A239567 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TriangularGridGraph.html">Triangular Grid Graph</a>
%e A239567 Triangle begins:
%e A239567    1;
%e A239567    3;
%e A239567    6,   6,    1;
%e A239567   10,  27,   21,    1;
%e A239567   15,  75,  151,  114,    27,     1;
%e A239567   21, 165,  615, 1137,   999,   353,   27;
%e A239567   28, 315, 1845, 6100, 11565, 12231, 6715, 1686, 150, 2;
%e A239567 ...
%e A239567 There is T(10, 19) = 1 way to place 19 points (X) on a grid of side 10 under to the condition mentioned above:
%e A239567                X
%e A239567               . .
%e A239567              . X .
%e A239567             X . . X
%e A239567            . . X . .
%e A239567           . X . . X .
%e A239567          X . . X . . X
%e A239567         . . X . . X . .
%e A239567        . X . . X . . X .
%e A239567       X . . X . . X . . X
%e A239567 This pattern seems to be the densest packing for all n == 1 (mod 3) and n >= 10.
%e A239567 From _Eric W. Weisstein_, Nov 11 2016: (Start)
%e A239567 Independence polynomials of the n-triangular grid graphs for n = 1, 2, ...:
%e A239567 1 + 3*x,
%e A239567 1 + 6*x + 6*x^2 + x^3,
%e A239567 1 + 10*x + 27*x^2 + 21*x^3 + x^4,
%e A239567 1 + 15*x + 75*x^2 + 151*x^3 + 114*x^4 + 27*x^5 + x^6,
%e A239567 ...
%e A239567 (End)
%Y A239567 Cf. A239438, A239572,
%Y A239567 Column 1 is A000217,
%Y A239567 Column 2 is A239568,
%Y A239567 Column 3 is A239569,
%Y A239567 Column 4 is A239570,
%Y A239567 Column 5 is A239571,
%Y A239567 Column 6 is A282998.
%Y A239567 Row sums are A027740(n)-1.
%K A239567 nonn,tabf
%O A239567 1,2
%A A239567 _Heinrich Ludwig_, Mar 21 2014