login
A301752
Clique covering number of the n-triangular grid graph.
1
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
Eric Weisstein's World of Mathematics, Clique Covering Number
Eric Weisstein's World of Mathematics, Triangular Grid Graph
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
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