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%I #55 Sep 06 2017 21:10:01
%S 1,1,3,4,6,7,10,12,15,19,22,26,31,35,40,46,51,57,64,70,77,85,92,100,
%T 109,117,126,136,145,155,166,176,187,199,210,222,235,247,260,274,287,
%U 301,316,330,345,361,376,392,409
%N Maximal number of points that can be placed on a triangular grid of side n so that there is no pair of adjacent points.
%C In other words, the independence number of the (n-1)-triangular grid graph.
%C Apart from a(3) and a(5) same as A007997(n+4) and A058212(n+2). - _Eric W. Weisstein_, Jun 14 2017
%C Also the independence number of the n-triangular honeycomb king graph. - _Eric W. Weisstein_, Sep 06 2017
%H Colin Barker, <a href="/A239438/b239438.txt">Table of n, a(n) for n = 1..1000</a>
%H A. V. Geramita, D. Gregory, and L. Roberts, <a href="http://dx.doi.org/10.1016/0022-4049(86)90029-0">Monomial ideals and points in projective space</a>, J. Pure Applied Alg 40 (1986), pp. 33-62.
%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="http://mathworld.wolfram.com/IndependenceNumber.html">Independence Number</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TriangularGridGraph.html">Triangular Grid Graph</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,1,-2,1).
%F a(n) = ceiling(n(n+1)/6) for n > 5, see Geramita, Gregory, & Roberts theorem 5.4. - _Charles R Greathouse IV_, Dec 04 2014
%F G.f.: x*(x^9-2*x^8+2*x^7-3*x^6+3*x^5-2*x^4+2*x^3-2*x^2+x-1) / ((x-1)^3*(x^2+x+1)). - _Colin Barker_, Feb 08 2015
%e On a triangular grid of side 5 at most a(5) = 6 points (X) can be placed so that there is no pair of adjacent points.
%e X
%e . .
%e X . X
%e . . . .
%e X . X . X
%t Table[1/18 (Piecewise[{{28, n == 2 || n == 4}}, 10] + 3 n (3 + n) + 8 Cos[(2 n Pi)/3]), {n, 0, 20}] (* _Eric W. Weisstein_, Jun 14 2017 *)
%o (PARI) Vec(x*(x^9-2*x^8+2*x^7-3*x^6+3*x^5-2*x^4+2*x^3-2*x^2+x-1)/((x-1)^3*(x^2+x+1)) + O(x^100)) \\ _Colin Barker_, Feb 08 2015
%Y Cf. A007997, A058212, A239567.
%K nonn,easy
%O 1,3
%A _Heinrich Ludwig_, Mar 18 2014
%E Extended by _Charles R Greathouse IV_, Dec 04 2014
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