A287195 Independence and clique covering number of the n-triangular honeycomb acute knight graph.

1, 3, 3, 5, 9, 9, 12, 18, 18, 22, 30, 30, 35, 45, 45, 51, 63, 63, 70, 84, 84, 92, 108, 108, 117, 135, 135, 145, 165, 165, 176, 198, 198, 210, 234, 234, 247, 273, 273, 287, 315, 315, 330, 360, 360, 376, 408, 408, 425, 459, 459, 477, 513, 513, 532, 570, 570

Offset

1,2

Comments

a(n) is also the length of row n in A244500.

Links

Colin Barker, 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, Independence Number.

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

Formula

From Colin Barker, Jul 15 2017: (Start)

G.f.: x*(1 + 2*x) / ((1 - x)^3*(1 + x + x^2)^2).

a(n) = a(n-1) + 2*a(n-3) - 2*a(n-4) - a(n-6) + a(n-7) for n>7.

(End)

Mathematica

LinearRecurrence[{1, 0, 2, -2, 0, -1, 1}, {1, 3, 3, 5, 9, 9, 12}, 50]
Table[1/18 ((n + 3) (3 n + 2) - 2 (n + 3) Cos[2 n Pi/3] - 2 Sqrt[3] (n + 1) Sin[2 n Pi/3]), {n, 50}]
Table[Piecewise[{{n (n + 3), Mod[n, 3] == 0}, {(n + 1) (n + 2), Mod[n, 3] == 1}, {(n + 1) (n + 4), Mod[n, 3] == 2}}]/6, {n, 50}]

Prog

(PARI) Vec(x*(1 + 2*x) / ((1 - x)^3*(1 + x + x^2)^2) + O(x^60)) \\ Colin Barker, Jul 15 2017

Crossrefs

Cf. A244500.

Sequence in context: A354953 A125960 A141584 * A179437 A136791 A213933

Adjacent sequences: A287192 A287193 A287194 * A287196 A287197 A287198

Keyword

nonn,easy

Author

Eric W. Weisstein, May 21 2017

Status

approved