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A290056
Number of cliques in the n-triangular graph.
3
1, 2, 8, 27, 76, 192, 456, 1045, 2344, 5186, 11364, 24719, 53444, 114948, 246096, 524713, 1114640, 2359942, 4981516, 10486691, 22021196, 46138632, 96470488, 201328317, 419432376, 872417482, 1811941876, 3758099255, 7784631444, 16106130956, 33286000544
OFFSET
1,2
LINKS
Eric Weisstein's World of Mathematics, Clique
Eric Weisstein's World of Mathematics, Johnson Graph
Eric Weisstein's World of Mathematics, Triangular Graph
FORMULA
a(n) = 1 + binomial(n,2) + (2^(n-1)-n)*n + binomial(n,3).
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
From Colin Barker, Jul 19 2017: (Start)
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).
a(n) = (6 + (-1+3*2^n)*n - 6*n^2 + n^3) / 6.
(End)
MATHEMATICA
Table[1 + Binomial[n, 2] + Binomial[n, 3] + (2^(n - 1) - n) n, {n, 20}] (* Eric W. Weisstein, Jul 19 2017 *)
LinearRecurrence[{8, -26, 44, -41, 20, -4}, {1, 2, 8, 27, 76, 192}, 20] (* Eric W. Weisstein, Jul 19 2017 *)
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 *)
PROG
(PARI) a(n) = 1 + binomial(n, 2) + (2^(n-1)-n)*n + binomial(n, 3);
(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
CROSSREFS
Cf. A000125 (maximal cliques), A000085 (independent vertex sets), A289837 (tetrahedral graph).
Sequence in context: A184628 A092071 A289864 * A100505 A102759 A292698
KEYWORD
nonn,easy
AUTHOR
Andrew Howroyd, Jul 19 2017
STATUS
approved