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A287195
Independence and clique covering number of the n-triangular honeycomb acute knight graph.
3
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
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.
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);
a(n) = A066377(n+1) - A092353(n). (End)
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
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
KEYWORD
nonn,easy
AUTHOR
Eric W. Weisstein, May 21 2017
STATUS
approved