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%I #7 Sep 18 2021 09:04:46
%S 1,4,10,25,64,163,415,1057,2692,6856,17461,44470,113257,288445,734617,
%T 1870936,4764934,12135421,30906712,78713779,200469691,510559873,
%U 1300303216,3311635996,8434135081,21480209374,54706189825,139326724105,354839847409,903712608748
%N Number of irredundant sets in the (2n-1)-triangular snake graph (for n > 1).
%C The 1-triangular snake is K_1, which has two trivial irredundant sets ({} and {1}), which differs from a(1).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IrredundantSet.html">Irredundant Set</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TriangularSnakeGraph.html">Triangular Snake Graph</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,1).
%F a(n) = 2*a(n-1)+a(n-2)+a(n-3) for n > 3.
%F G.f.: x*(-1-2*x-x^2)/(-1+2*x+x^2+x^3).
%t Table[-RootSum[-1 - # - 2 #^2 + #^3 &, -9 #^n - 16 #^(n + 1) + 5 #^(n + 2) &]/29, {n, 20}]
%t LinearRecurrence[{2, 1, 1}, {1, 4, 10}, 20]
%t CoefficientList[Series[(-1 - 2 x - x^2)/(-1 + 2 x + x^2 + x^3), {x, 0, 20}], x]
%K nonn,easy
%O 1,2
%A _Eric W. Weisstein_, Sep 11 2021