A232179 Least k >= 0 such that n^2 + triangular(k) is a triangular number.

0, 0, 3, 1, 15, 2, 0, 3, 63, 4, 8, 5, 11, 6, 20, 3, 255, 8, 1, 9, 3, 10, 38, 11, 59, 12, 45, 13, 8, 14, 2, 15, 1023, 16, 59, 0, 24, 18, 66, 19, 51, 20, 3, 21, 44, 10, 80, 23, 251, 24, 42, 25, 68, 26, 4, 27, 39, 28, 101, 29, 10, 30, 108, 8, 4095, 32, 5, 33, 128

Offset: 0

Author

Alex Ratushnyak, Nov 20 2013

Keywords

• nonn

Comments

Triangular(k) = k*(k+1)/2.

Links

• Chai Wah Wu, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A000217, A000290, A001109.

Cf. A082183 (least k>0 such that triangular(n) + triangular(k) is a triangular number).

Cf. A232177 (least k>0 such that triangular(n) + triangular(k) is a square).

Cf. A232176 (least k>0 such that n^2 + triangular(k) is a square).

Cf. A101157 (least k>0 such that triangular(n) + k^2 is a triangular number).

Cf. A232178 (least k>=0 such that triangular(n) + k^2 is a square).

Programs

Mathematica

TriangularQ[n_] := IntegerQ[Sqrt[1 + 8*n]]; Table[k = 0; While[! TriangularQ[n^2 + k*(k + 1)/2], k++]; k, {n, 0, 68}] (* T. D. Noe, Nov 21 2013 *)

PARI

a(n) = {my(k = 0); while (! ispolygonal(n^2 + k*(k+1)/2, 3), k++); k;} \\ Michel Marcus, Sep 15 2017

Python

from _future_ import division
from sympy import divisors
def A232179(n):
    if n == 0:
        return 0
    t = 2*n**2
    ds = divisors(t)
    for i in range(len(ds)//2-1,-1,-1):
        x = ds[i]
        y = t//x
        a, b = divmod(y-x,2)
        if b:
            return a
    return -1 # Chai Wah Wu, Sep 12 2017

Formula

a(A001109(n)) = 0.