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

1,3

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

Links

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

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

Example

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).

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 *)

Prog

(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

Crossrefs

Cf. A067721, A115878, A115879, A115881.

Sequence in context: A290826 A124228 A115879 * A335728 A078037 A088508

Adjacent sequences: A115877 A115878 A115879 * A115881 A115882 A115883

Keyword

nonn

Author

Giovanni Resta, Feb 02 2006

Status

approved