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Number of (not necessarily maximal) cliques in the n X n black bishop graph.
1

%I #6 Dec 28 2022 10:35:15

%S 2,4,14,30,82,160,386,718,1646,3000,6742,12190,27194,49024,109082,

%T 196446,436726,786232,1747406,3145486,6990242,12582624,27961714,

%U 50331310,111847742,201326200,447392006,805305918,1789569226,3221224960,7158278282,12884901310

%N Number of (not necessarily maximal) cliques in the n X n black bishop graph.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BlackBishopGraph.html">Black Bishop Graph</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Clique.html">Clique</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2, 4, -10, 1, 8, -4).

%F a(n) = ((-2)^(n + 1) + (-1)^n + 19*2^(n + 1) - 6*n*(n + 4) - 25)/12.

%F a(n) = 2*a(n-1) + 4*a(n-2) - 10*a(n-3) + a(n-4) + 8*a(n-5) - 4*a(n-6).

%F G.f.: -2*x*(-1 + x^2 - 3*x^3 - 2*x^4 + 2*x^5)/((-1 + x)^3 (-1 - x + 4*x^2 + 4*x^3)).

%t Table[((-2)^(n + 1) + (-1)^n + 19 2^(n + 1) - 6 n (n + 4) - 25)/12, {n, 20}]

%t LinearRecurrence[{2, 4, -10, 1, 8, -4}, {2, 4, 14, 30, 82, 160}, 20]

%t CoefficientList[Series[-2 (-1 + x^2 - 3 x^3 - 2 x^4 + 2 x^5)/((-1 + x)^3 (-1 - x + 4 x^2 + 4 x^3)), {x, 0, 20}], x]

%K nonn

%O 1,1

%A _Eric W. Weisstein_, Nov 29 2017