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A290056 Number of cliques in the n-triangular graph.

%I A290056 #12 Feb 16 2025 08:33:49
%S A290056 1,2,8,27,76,192,456,1045,2344,5186,11364,24719,53444,114948,246096,
%T A290056 524713,1114640,2359942,4981516,10486691,22021196,46138632,96470488,
%U A290056 201328317,419432376,872417482,1811941876,3758099255,7784631444,16106130956,33286000544
%N A290056 Number of cliques in the n-triangular graph.
%H A290056 Colin Barker, <a href="/A290056/b290056.txt">Table of n, a(n) for n = 1..1000</a>
%H A290056 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Clique.html">Clique</a>
%H A290056 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/JohnsonGraph.html">Johnson Graph</a>
%H A290056 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TriangularGraph.html">Triangular Graph</a>
%H A290056 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (8,-26,44,-41,20,-4).
%F A290056 a(n) = 1 + binomial(n,2) + (2^(n-1)-n)*n + binomial(n,3).
%F A290056 a(n) = 8*a(n-1)-26*a(n-2)+44*a(n-3)-41*a(n-4)+20*a(n-5)-4*a(n-6). - _Eric W. Weisstein_, Jul 29 2017
%F A290056 From _Colin Barker_, Jul 19 2017: (Start)
%F A290056 G.f.: x*(1 - 6*x + 18*x^2 - 29*x^3 + 21*x^4 - 4*x^5) / ((1 - x)^4*(1 - 2*x)^2).
%F A290056 a(n) = (6 + (-1+3*2^n)*n - 6*n^2 + n^3) / 6.
%F A290056 (End)
%t A290056 Table[1 + Binomial[n, 2] + Binomial[n, 3] + (2^(n - 1) - n) n, {n, 20}] (* _Eric W. Weisstein_, Jul 19 2017 *)
%t A290056 LinearRecurrence[{8, -26, 44, -41, 20, -4}, {1, 2, 8, 27, 76, 192}, 20] (* _Eric W. Weisstein_, Jul 19 2017 *)
%t A290056 CoefficientList[Series[(1 - 6 x + 18 x^2 - 29 x^3 + 21 x^4 - 4 x^5)/((-1 + x)^4 (-1 + 2 x)^2), {x, 0, 20}], x] (* _Eric W. Weisstein_, Jul 19 2017 *)
%o A290056 (PARI) a(n) = 1 + binomial(n,2) + (2^(n-1)-n)*n + binomial(n,3);
%o A290056 (PARI) Vec(x*(1 - 6*x + 18*x^2 - 29*x^3 + 21*x^4 - 4*x^5) / ((1 - x)^4*(1 - 2*x)^2) + O(x^40)) \\ _Colin Barker_, Jul 19 2017
%Y A290056 Cf. A000125 (maximal cliques), A000085 (independent vertex sets), A289837 (tetrahedral graph).
%K A290056 nonn,easy
%O A290056 1,2
%A A290056 _Andrew Howroyd_, Jul 19 2017