cp's OEIS Frontend

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

%I A287195 #37 Feb 16 2025 08:33:46
%S A287195 1,3,3,5,9,9,12,18,18,22,30,30,35,45,45,51,63,63,70,84,84,92,108,108,
%T A287195 117,135,135,145,165,165,176,198,198,210,234,234,247,273,273,287,315,
%U A287195 315,330,360,360,376,408,408,425,459,459,477,513,513,532,570,570
%N A287195 Independence and clique covering number of the n-triangular honeycomb acute knight graph.
%C A287195 a(n) is also the length of row n in A244500.
%H A287195 Colin Barker, <a href="/A287195/b287195.txt">Table of n, a(n) for n = 1..1000</a>
%H A287195 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 A287195 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CliqueCoveringNumber.html">Clique Covering Number</a>.
%H A287195 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/IndependenceNumber.html">Independence Number</a>.
%H A287195 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,2,-2,0,-1,1).
%F A287195 From _Colin Barker_, Jul 15 2017: (Start)
%F A287195 G.f.: x*(1 + 2*x) / ((1 - x)^3*(1 + x + x^2)^2).
%F A287195 a(n) = a(n-1) + 2*a(n-3) - 2*a(n-4) - a(n-6) + a(n-7) for n>7. (End)
%F A287195 From _Ridouane Oudra_, Jun 23 2024: (Start)
%F A287195 a(n) = Sum_{i=1..n+3} (i mod 3)*floor(i/3);
%F A287195 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);
%F A287195 a(n) = A066377(n+1) - A092353(n). (End)
%F A287195 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
%t A287195 LinearRecurrence[{1, 0, 2, -2, 0, -1, 1}, {1, 3, 3, 5, 9, 9, 12}, 50]
%t A287195 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}]
%t A287195 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}]
%o A287195 (PARI) Vec(x*(1 + 2*x) / ((1 - x)^3*(1 + x + x^2)^2) + O(x^60)) \\ _Colin Barker_, Jul 15 2017
%Y A287195 Cf. A244500.
%Y A287195 Cf. A066377, A092353.
%K A287195 nonn,easy
%O A287195 1,2
%A A287195 _Eric W. Weisstein_, May 21 2017