A115880 Largest positive x satisfying the Diophantine equation x^2 = y*(y+n), a(n)=0 if there are no solutions.

0, 0, 2, 0, 6, 4, 12, 3, 20, 12, 30, 8, 42, 24, 56, 15, 72, 40, 90, 24, 110, 60, 132, 35, 156, 84, 182, 48, 210, 112, 240, 63, 272, 144, 306, 80, 342, 180, 380, 99, 420, 220, 462, 120, 506, 264, 552, 143, 600, 312, 650, 168, 702, 364, 756, 195, 812, 420, 870

Offset: 1

Author

Giovanni Resta, Feb 02 2006

Keywords

• nonn

Comments

Notice that x^2 = y*(y+n) is equivalent to (n+2*y+2*x)*(n+2*y-2*x) = n^2. We take the factorization of n^2 into two factors congruent mod 4 where one is as small as possible and the other is as large as possible. For n == 0 mod 4 the factors are 4 and n^2/4, for n == 2 mod 4 they are 2 and n^2/2, for n odd they are 1 and n^2. - Robert Israel, Jun 27 2014

Examples

			a(15)=56 since the solutions (x,y) to x^2=y(y+15) are (4,1), (10,5), (18, 12) and (56, 49). The largest x is 56, from (x,y)=(56,49).
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A067721, A115878, A115879, A115881.

Programs

Mathematica

Table[Max[x/.Solve[{x^2==y(y+n),x>0},{x,y},Integers]],{n,1,100}]/.x->0 (* Vaclav Kotesovec, Jun 26 2014 *)

Python

def A115880(n):
    a, b = divmod(n,4)
    return (a**2-1,(c:=a<<1)*(c+1),c*(a+1),c*(c+3)+2)[b] # Chai Wah Wu, Aug 21 2024

Formula

Empirical g.f.: x^3*(x^9-2*x^6-3*x^5-6*x^4-4*x^3-6*x^2-2) / ((x-1)^3*(x+1)^3*(x^2+1)^3). - Colin Barker, Jun 26 2014

a(4*j) = j^2 - 1,

a(4*j+1) = 4*j^2+2*j,

a(4*j+2) = 2*j^2+2*j,

a(4*j+3) = 4*j^2+6*j+2. (see Comments) - Robert Israel, Jun 27 2014