login
Clique covering number of the n-triangular grid graph.
1

%I #23 Dec 13 2024 09:39:36

%S 1,3,4,6,8,10,13,15,19,22,26,31,35,41,46,52,58,64,71,77,85,92,100,109,

%T 117,127,136,146,156,166,177,187,199,210,222,235,247,261,274,288,302,

%U 316,331,345,361,376,392,409,425,443,460,478,496,514,533,551,571

%N Clique covering number of the n-triangular grid graph.

%C Maximal cliques are triangles in the n-triangular grid graph. The clique covering number cannot be less than the number of nodes divided by three. Perfect nonoverlapping coverings are possible for n + 1 in A072065. - _Andrew Howroyd_, Jun 27 2018

%H Georg Fischer, <a href="/A301752/b301752.txt">Table of n, a(n) for n = 1..1000</a>

%H 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 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CliqueCoveringNumber.html">Clique Covering Number</a>.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TriangularGridGraph.html">Triangular Grid Graph</a>.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TriangularHoneycombKingGraph.html">Triangular Honeycomb King Graph</a>.

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-1,-2,2,1,0,-2,1).

%F a(n) ~ (n+1)*(n+2)/6. - _Andrew Howroyd_, Jun 27 2018

%F a(n) = 2*a(n-1) - a(n-3) - 2*a(n-4) + 2*a(n-5) + a(n-6) - 2*a(n-8) + a(n-9). - _Eric W. Weisstein_, Apr 18 2019

%F G.f.: x (-1 - x + 2*x^2 + x^3 - x^4 - 2*x^5 + 2*x^7 - x^8)/((-1 + x)^3*(1 + x - x^3 + x^5 + x^6)). - _Eric W. Weisstein_, Apr 18 2019

%t Table[(Sqrt[3] (16 + 3 n (3 + n)) - 9 Cos[n Pi/6] + 2 Sqrt[3] Cos[2 n Pi/3] + 9 Cos[5 n Pi/6] + 9 Sin[n Pi/6] - 9 Sin[5 n Pi/6])/(18 Sqrt[3]), {n, 20}] (* _Eric W. Weisstein_, Apr 18 2019 *)

%t LinearRecurrence[{2, 0, -1, -2, 2, 1, 0, -2, 1}, {1, 3, 4, 6, 8, 10, 13, 15, 19}, 20] (* _Eric W. Weisstein_, Apr 18 2019 *)

%t CoefficientList[Series[(-1 - x + 2 x^2 + x^3 - x^4 - 2 x^5 + 2 x^7 - x^8)/((-1 + x)^3 (1 + x - x^3 + x^5 + x^6)), {x, 0, 20}], x] (* _Eric W. Weisstein_, Apr 18 2019 *)

%Y Cf. A072065.

%K nonn

%O 1,2

%A _Eric W. Weisstein_, Mar 26 2018

%E a(11)-a(24) from _Andrew Howroyd_, Jun 27 2018

%E More terms from _Georg Fischer_, Jun 04 2019