A335640 - OEIS

%I #31 Apr 04 2023 22:10:48

%S 4,9,16,25,36,45,49,64,81,96,100,121,144,169,175,196,225,256,288,289,

%T 320,324,361,400,441,484,529,576,625,640,676,729,784,841,891,900,961,

%U 1024,1089,1156,1200,1225,1296,1350,1369,1444,1521,1573,1600,1681,1764,1849,1936,2016

%N 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.

%C From _Robert Israel_, Apr 03 2023: (Start)

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

%C 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.

%C 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.

%C (End)

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

%p N:= 3000: # for terms <= N

%p R:= {4}:

%p for t from 2 to N/2 do

%p   for r from t+1 to N/2 do

%p     c:= r^2-r;

%p     b:= 1+t*r;

%p     delta:= b^2 - 4*c;

%p     if not issqr(delta) then next fi;

%p     delta:= sqrt(delta);

%p     S:= select(x -> x::posint and x >= r and r+x <= N, {(b+delta)/2,(b-delta)/2});

%p     R:= R union map(`+`,S,r);

%p od od:

%p sort(convert(R,list)); # _Robert Israel_, Apr 04 2023

%t 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

%Y Cf. A000290, A152618.

%K nonn

%O 1,1

%A _Wesley Ivan Hurt_, Oct 04 2020