A335640 Numbers k of the form r^2 - t*r*s + s^2, where r, s and t are positive integers, r + s = k and t < r <= s.

4, 9, 16, 25, 36, 45, 49, 64, 81, 96, 100, 121, 144, 169, 175, 196, 225, 256, 288, 289, 320, 324, 361, 400, 441, 484, 529, 576, 625, 640, 676, 729, 784, 841, 891, 900, 961, 1024, 1089, 1156, 1200, 1225, 1296, 1350, 1369, 1444, 1521, 1573, 1600, 1681, 1764, 1849, 1936, 2016

Offset

1,1

Comments

From Robert Israel, Apr 03 2023: (Start)

Includes m^2 for m >= 2: for k = m^2. take t = 2, r = (m^2 - m)/2, s = (m^2 + m)/2.

Includes A152618(n) = (n-1)^2*(n+1) for n >= 3: take t = n - 1, r = n^2 - n, s = n^3 - 2*n^2 + 1.

Another infinite family of solutions: t = 3, r = y - 1, s = (x + 3*y)/2 - 1, k = (x + 5*y)/2 - 2 where x and y satisfy the Pell-type equation x^2 + 4 = 5*y^2.

(End)

Links

Table of n, a(n) for n=1..54.

Example

9 is in the sequence since 9 = 3^2 - 2*3*6 + 6^2.

Maple

N:= 3000: # for terms <= N
R:= {4}:
for t from 2 to N/2 do
  for r from t+1 to N/2 do
    c:= r^2-r;
    b:= 1+t*r;
    delta:= b^2 - 4*c;
    if not issqr(delta) then next fi;
    delta:= sqrt(delta);
    S:= select(x -> x::posint and x >= r and r+x <= N, {(b+delta)/2, (b-delta)/2});
    R:= R union map(`+`, S, r);
od od:
sort(convert(R, list)); # Robert Israel, Apr 04 2023

Mathematica

Table[If[Sum[Sum[KroneckerDelta[i^2 - k*i (n - i) + (n - i)^2, n], {k, i - 1}], {i, Floor[n/2]}] > 0, n, {}], {n, 200}] // Flatten

Crossrefs

Cf. A000290, A152618.

Sequence in context: A070459 A292676 A241971 * A302053 A162496 A159852

Adjacent sequences: A335637 A335638 A335639 * A335641 A335642 A335643

Keyword

nonn

Author

Wesley Ivan Hurt, Oct 04 2020

Status

approved