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A357285 a(n) = number of subsets S of {1,2,...,n} having more than 2 elements such that (sum of least three elements of S) < max(S).

%I A357285 #4 Oct 02 2022 13:34:06
%S A357285 0,0,0,0,0,0,0,8,32,104,304,792,1920,4520,10192,22392,48416,102856,
%T A357285 215664,448792,925632,1897064,3872016,7868344,15936096,32208136,
%U A357285 64946096,130738776,262886656,527990696,1059498576,2124829944,4258791328,8532044360,17087943920
%N A357285 a(n) = number of subsets S of {1,2,...,n} having more than 2 elements such that (sum of least three elements of S) < max(S).
%H A357285 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (4, -3, -2, -6, 8, 24, -40, 16).
%F A357285 a(n) = 4*a(n-1) - 3*a(n-2) - 2*a(n-3) - 6*a(n-4) + 8*a(n-5) + 24*a(n-6) - 40*a(n-7).
%F A357285 G.f.: -((8 x^7)/((-1 + x)^2 (-1 + 2 x) (-1 + 2 x^2) (-1 + 4 x^3))).
%e A357285 The 8 relevant subsets of {1,2,3,4,5,6,7} are {1,2,3,7}, {1,2,3,4,7}, {1,2,3,5,7}, {1,2,3,6,7}, {1,2,3,4,5,7}, {1,2,3,4,6,7}, {1,2,3,5,6,7}, and {1,2,3,4,5,6,7}.
%t A357285 s[n_] := s[n] = Select[Subsets[Range[n]], Length[#] >= 3 &];
%t A357285 a[n_] := Select[s[n], #[[1]] + #[[2]] + #[[3]] < #[[-1]] &]
%t A357285 Table[Length[a[n]], {n, 0, 18}]
%Y A357285 Cf. A357286, A357287, A357289.
%K A357285 nonn,easy
%O A357285 0,8
%A A357285 _Clark Kimberling_, Oct 02 2022