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A239438
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Maximal number of points that can be placed on a triangular grid of side n so that there is no pair of adjacent points.
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5
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1, 1, 3, 4, 6, 7, 10, 12, 15, 19, 22, 26, 31, 35, 40, 46, 51, 57, 64, 70, 77, 85, 92, 100, 109, 117, 126, 136, 145, 155, 166, 176, 187, 199, 210, 222, 235, 247, 260, 274, 287, 301, 316, 330, 345, 361, 376, 392, 409
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OFFSET
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1,3
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COMMENTS
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In other words, the independence number of the (n-1)-triangular grid graph.
Also the independence number of the n-triangular honeycomb king graph. - Eric W. Weisstein, Sep 06 2017
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LINKS
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FORMULA
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a(n) = ceiling(n(n+1)/6) for n > 5, see Geramita, Gregory, & Roberts theorem 5.4. - Charles R Greathouse IV, Dec 04 2014
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
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EXAMPLE
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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.
X
. .
X . X
. . . .
X . X . X
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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