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A385879 Values of u in triples (u, v, w) such that the polynomial x^3 + u*x^2 + v*x + w has 3 (possibly repeated) negative integer zeros; the triples are ordered by the inequality u < v.

%I A385879 #14 Jul 27 2025 21:57:27
%S A385879 3,4,5,5,6,6,6,7,7,7,7,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,
%T A385879 10,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,
%U A385879 13,13,13,13,13,13,13,13,13,13,13,13,13,13,14,14,14
%N A385879 Values of u in triples (u, v, w) such that the polynomial x^3 + u*x^2 + v*x + w has 3 (possibly repeated) negative integer zeros; the triples are ordered by the inequality u < v.
%e A385879 First 20 triples:
%e A385879     u    v    w
%e A385879     3    3    1
%e A385879     4    5    2
%e A385879     5    7    3
%e A385879     5    8    4
%e A385879     6    9    4
%e A385879     6   11    6
%e A385879     6   12    8
%e A385879     7   11    5
%e A385879     7   14    8
%e A385879     7   15    9
%e A385879     7   16   12
%e A385879     8   13    6
%e A385879     8   17   10
%e A385879     8   19   12
%e A385879     8   20   16
%e A385879     8   21   18
%e A385879     9   15    7
%e A385879     9   20   12
%e A385879     9   23   15
%e A385879     9   24   16
%e A385879 (x + 1)^3 = x^3 + 3*x^2 + 3*x + 1, so (3, 3, 1) is in the list; here the negative zeros are -1, -1, and -1.
%t A385879 z = 120;
%t A385879 t = Table[{b + c + d, c  d + d  b + b  c, b  c  d}, {b, 1, z}, {c, 1, z}, {d, 1, z}];
%t A385879 t1 = Union[Flatten[t, 2]]; t2 = Take[t1, 40]
%t A385879 Grid[t2]
%Y A385879 Cf. A385880, A386285, A386286.
%K A385879 nonn
%O A385879 1,1
%A A385879 _Clark Kimberling_, Jul 11 2025