A393713 a(n) = number of triples (x, y, z) such that x^2 + y*z = n, where x, y, z are positive integers satisfying y < x < z.

0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 2, 3, 2, 4, 3, 4, 3, 4, 4, 4, 4, 4, 4, 6, 5, 5, 6, 5, 5, 7, 5, 6, 6, 6, 5, 9, 6, 7, 8, 7, 6, 10, 7, 7, 9, 8, 7, 9, 7, 10, 10, 8, 9, 11, 8, 9, 11, 10, 9, 11, 9, 11, 10, 10, 10, 14, 9, 11, 14, 11, 10, 13

Offset

0,14

Links

Robert Israel, Table of n, a(n) for n = 0..10000

Example

a(17) = 3 counts these triples: (2, 1, 13), (3, 1, 8), (3, 2, 4).

Maple

N:= 100: # for a(0) .. a(N)
V:= Array(0..N):
for y from 1 while (y+1)^2 + y*(y+2) <= N do
  for x from y+1 while x^2 + y*(x+1) <= N do
    for z from x+1 do
      v:= x^2 + y*z;
      if v > N then break fi;
      V[v]:= V[v]+1
od od od:
convert(V, list); # Robert Israel, Apr 13 2026

Mathematica

t[n_, c_] := Module[{r}, r = Flatten[Table[If[n - x^2 <= 0, {},
       Map[({x, #, Quotient[n - x^2, #]} &),
       Select[Divisors[n - x^2], Divisible[n - x^2, #] &]]], {x, 1,
       Floor[Sqrt[n - 1]]}], 1]; Select[r, Apply[c, #] &]];
c = (#2 < #1 && #1 < #3 &); Table[{n, t[n, c]}, {n, 1, 30}]
Join[{0}, Table[Length[t[n, c]], {n, 1, 90}]]
(* Peter J. C. Moses, Mar 29 2026 *)

Crossrefs

Cf. A393710, A394746.

Sequence in context: A216325 A322868 A240975 * A242166 A068211 A236832

Adjacent sequences: A393710 A393711 A393712 * A393714 A393715 A393716

Keyword

nonn

Author

Clark Kimberling, Apr 01 2026

Status

approved