A115881 a(n) is the largest positive y satisfying the Diophantine equation x^2=y(y+n). a(n)=0 if there are no solutions.

0, 0, 1, 0, 4, 2, 9, 1, 16, 8, 25, 4, 36, 18, 49, 9, 64, 32, 81, 16, 100, 50, 121, 25, 144, 72, 169, 36, 196, 98, 225, 49, 256, 128, 289, 64, 324, 162, 361, 81, 400, 200, 441, 100, 484, 242, 529, 121, 576, 288, 625, 144, 676, 338, 729, 169, 784, 392, 841, 196

Offset

1,5

Comments

The corresponding least y is given by A067721(n).

Links

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

Formula

Empirical g.f.: -x^3*(x^9+x^8+2*x^7+4*x^6+x^5+6*x^4+2*x^3+4*x^2+1) / ((x-1)^3*(x+1)^3*(x^2+1)^3). - Colin Barker, Jun 26 2014

From empirical g.f.: a(n) = 1/2 - n/2 + 11*n^2/64 + (1/4 - 1/32*n^2) * (2*floor(n/4) + 2*floor((n+1)/4) - n + 1) + (1/4 - 5/64*n^2)*(-1)^n. - Vaclav Kotesovec, Jun 26 2014

From Chai Wah Wu, Aug 21 2024: (Start)

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

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

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

a(4*j+3) = 4*j^2+4*j+1 (see A115880).

(End)

Example

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

Mathematica

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

Prog

(Python)
def A115881(n):
    a, b = divmod(n, 4)
    return ((c:=a**2)-(a<<1)+1, (d:=c<<2), c<<1, d+(a<<2)+1)[b] # Chai Wah Wu, Aug 21 2024

Crossrefs

Cf. A067721, A115878, A115879, A115880.

Sequence in context: A322550 A201531 A021237 * A180434 A201574 A077809

Adjacent sequences: A115878 A115879 A115880 * A115882 A115883 A115884

Keyword

nonn

Author

Giovanni Resta, Feb 02 2006

Status

approved