OFFSET
1,2
COMMENTS
A 4-tuple (m, u, v, w) is a quartet of type 3 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 = 4.
EXAMPLE
First 20 quartets (4,u,v,w) of type 3:
m u v w
4 1 12 11
4 2 8 7
4 6 8 9
4 7 6 8
4 7 12 13
4 8 16 17
4 9 10 12
4 9 20 21
4 10 8 11
4 10 12 14
4 10 24 25
4 11 14 16
4 11 28 29
4 12 16 18
4 12 32 33
4 13 6 12
4 13 12 15
4 13 18 20
4 13 36 37
4 14 5 13
4*(4-7) = 6*(6-8), so (4,7,6,8) is in the list.
MATHEMATICA
solnsM[m_, max_] := Module[{ans = {}, rhs = {}, u, v, w, lhs, matching},
Do[Do[AppendTo[rhs, {v*(v - w), v, w}], {w, max}], {v, m*(m + max)}];
rhs = GatherBy[rhs, First];
Do[lhs = m*(m - u); matching = Select[rhs, #[[1, 1]] == lhs &];
If[Length[matching] > 0, Do[AppendTo[ans,
Map[{m, u, #[[2]], #[[3]]} &, matching[[1]]]], {i,
Length[matching]}]], {u, max}];
ans = Flatten[ans, 1];
Select[Union[Map[Sort[{#, RotateLeft[#, 2]}][[1]] &,
Sort[Select[DeleteDuplicates[ans],
Length[Union[#]] == 4 &]]]], #[[1]] == m &]];
TableForm[solns = solnsM[4, 100],
TableHeadings -> {None, {"m", "u", "v", "w"}}]
Map[#[[2]] &, solns] (* u, A390024 *)
Map[#[[3]] &, solns] (* v, A390025 *)
Map[#[[4]] &, solns] (* w, A390026 *)
(* Peter J. C. Moses, Jun 15 2025 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, Oct 29 2025
STATUS
approved
