A385884 - OEIS

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.

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, 18, 18, 19, 19, 19, 20, 20, 21, 22, 22, 22, 23, 23, 24, 24, 25, 25, 25, 26, 26, 26, 27, 28, 28, 28, 28, 28, 29, 30, 30, 31, 31, 31, 32, 32, 33, 33, 33, 34, 34, 34

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OFFSET

1,2

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..67.

EXAMPLE

First 20 quartets (2,u,v,w) of type 2:

m u v w

2 1 6 5

2 3 10 9

2 4 12 11

2 5 14 13

2 6 16 15

2 7 6 3

2 7 18 17

2 8 5 1

2 8 20 19

2 9 22 21

2 10 8 5

2 10 24 23

2 11 26 25

2 12 7 3

2 12 28 27

2 13 6 1

2 13 10 7

2 13 30 29

2 14 8 4

2 14 32 31

2 (2 +4) = 12 (12 - 11), so (2,4,12,11) 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[2, 100],

TableHeadings -> {None, {"m", "u", "v", "w"}}]

u1 = Map[#[[2]] &, solns] (*u, A385884 *)

v1 = Map[#[[3]] &, solns] (*v, A386216 *)

w1 = Map[#[[4]] &, solns] (*w, A386982 *)

(* Peter J. C. Moses, Jun 15 2025 *)

CROSSREFS

Cf. A385182 (type 1, m=1), A386630 (type 3, m=1).

Sequence in context: A121857 A121854 A196119 * A225852 A198458 A134483

Adjacent sequences: A385881 A385882 A385883 * A385885 A385886 A385887

KEYWORD

nonn

AUTHOR

Clark Kimberling, Aug 07 2025

STATUS

approved