A301752 Clique covering number of the n-triangular grid graph.

1, 3, 4, 6, 8, 10, 13, 15, 19, 22, 26, 31, 35, 41, 46, 52, 58, 64, 71, 77, 85, 92, 100, 109, 117, 127, 136, 146, 156, 166, 177, 187, 199, 210, 222, 235, 247, 261, 274, 288, 302, 316, 331, 345, 361, 376, 392, 409, 425, 443, 460, 478, 496, 514, 533, 551, 571

Offset

1,2

Comments

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

Links

Georg Fischer, Table of n, a(n) for n = 1..1000

Stan Wagon, Graph Theory Problems from Hexagonal and Traditional Chess., The College Mathematics Journal, Vol. 45, No. 4, September 2014, pp. 278-287.

Eric Weisstein's World of Mathematics, Clique Covering Number.

Eric Weisstein's World of Mathematics, Triangular Grid Graph.

Eric Weisstein's World of Mathematics, Triangular Honeycomb King Graph.

Index entries for linear recurrences with constant coefficients, signature (2,0,-1,-2,2,1,0,-2,1).

Formula

a(n) ~ (n+1)*(n+2)/6. - Andrew Howroyd, Jun 27 2018

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

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

Mathematica

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 *)
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 *)
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 *)

Crossrefs

Cf. A072065.

Sequence in context: A247422 A256698 A024182 * A173339 A064269 A260485

Adjacent sequences: A301749 A301750 A301751 * A301753 A301754 A301755

Keyword

nonn

Author

Eric W. Weisstein, Mar 26 2018

Extensions

a(11)-a(24) from Andrew Howroyd, Jun 27 2018

More terms from Georg Fischer, Jun 04 2019

Status

approved