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
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).
Programs
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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}] -
PARI
Vec(x*(1 + 2*x) / ((1 - x)^3*(1 + x + x^2)^2) + O(x^60)) \\ Colin Barker, Jul 15 2017
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)
From Ridouane Oudra, Jun 23 2024: (Start)
a(n) = Sum_{i=1..n+3} (i mod 3)*floor(i/3);
a(n) = (1/2)*(n^2 + n + (n^2 - 5*n)*t -(6*n - 9)*t^2 + 9*t^3), where t = floor(n/3);
E.g.f.: exp(-x/2)*(exp(3*x/2)*(6 + 14*x + 3*x^2) - 2*(3 + x)*cos(sqrt(3)*x/2) - 2*sqrt(3)*(1 - x)*sin(sqrt(3)*x/2))/18. - Stefano Spezia, Jun 23 2024
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