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A385884 Values of u in the quartets (2, u, v, w) of type 2; i.e., values of u for solutions to 2*(2 + u) = v*(v - w), in positive integers, with v > 1, sorted by nondecreasing values of u; see Comments.

%I A385884 #10 Aug 16 2025 16:07:05
%S A385884 1,3,4,5,6,7,7,8,8,9,10,10,11,12,12,13,13,13,14,14,15,16,16,16,17,18,
%T A385884 18,18,19,19,19,20,20,21,22,22,22,23,23,24,24,25,25,25,26,26,26,27,28,
%U A385884 28,28,28,28,29,30,30,31,31,31,32,32,33,33,33,34,34,34
%N A385884 Values of u in the quartets (2, u, v, w) of type 2; i.e., values of u for solutions to 2*(2 + u) = v*(v - w), in positive integers, with v > 1, sorted by nondecreasing values of u; see Comments.
%C A385884 A 4-tuple (m, u, v, w) is a quartet of type 2 if m, u, v, w are distinct positive integers such that m < v and m*(m + u) = v*(v - w). Here, the values of u are arranged in nondecreasing order. When there is more than one solution for given m and u, the values of v are arranged in increasing order. Here, m = 2.
%e A385884 First 20 quartets (2,u,v,w) of type 2:
%e A385884   m   u   v   w
%e A385884   2   1   6   5
%e A385884   2   3  10   9
%e A385884   2   4  12  11
%e A385884   2   5  14  13
%e A385884   2   6  16  15
%e A385884   2   7   6   3
%e A385884   2   7  18  17
%e A385884   2   8   5   1
%e A385884   2   8  20  19
%e A385884   2   9  22  21
%e A385884   2  10   8   5
%e A385884   2  10  24  23
%e A385884   2  11  26  25
%e A385884   2  12   7   3
%e A385884   2  12  28  27
%e A385884   2  13   6   1
%e A385884   2  13  10   7
%e A385884   2  13  30  29
%e A385884   2  14   8   4
%e A385884   2  14  32  31
%e A385884 2 (2 +4) = 12 (12 - 11), so (2,4,12,11) is in the list.
%t A385884 solnsM[m_, max_] :=
%t A385884   Module[{ans = {}, rhs = {}, u, v, w, lhs, matching},
%t A385884    Do[Do[AppendTo[rhs, {v*(v - w), v, w}], {w, max}], {v,
%t A385884      m*(m + max)}];
%t A385884    rhs = GatherBy[rhs, First];
%t A385884    Do[lhs = m*(m + u); matching = Select[rhs, #[[1, 1]] == lhs &];
%t A385884     If[Length[matching] > 0,
%t A385884      Do[AppendTo[ans,
%t A385884        Map[{m, u, #[[2]], #[[3]]} &, matching[[1]]]], {i,
%t A385884        Length[matching]}]], {u, max}];
%t A385884    ans = Flatten[ans, 1];
%t A385884    Select[
%t A385884     Union[Map[Sort[{#, RotateLeft[#, 2]}][[1]] &,
%t A385884       Sort[Select[DeleteDuplicates[ans],
%t A385884         Length[Union[#]] == 4 &]]]], #[[1]] == m &]];
%t A385884 TableForm[solns = solnsM[2, 100],
%t A385884  TableHeadings -> {None, {"m", "u", "v", "w"}}]
%t A385884 u1 = Map[#[[2]] &, solns]  (*u, A385884 *)
%t A385884 v1 = Map[#[[3]] &, solns]  (*v, A386216 *)
%t A385884 w1 = Map[#[[4]] &, solns]  (*w, A386982 *)
%t A385884 (* _Peter J. C. Moses_, Jun 15 2025 *)
%Y A385884 Cf. A385182 (type 1, m=1), A386630 (type 3, m=1).
%K A385884 nonn
%O A385884 1,2
%A A385884 _Clark Kimberling_, Aug 07 2025