A232176 Least positive k such that n^2 + triangular(k) is a square.

1, 2, 6, 10, 14, 18, 7, 5, 8, 34, 6, 42, 46, 15, 54, 16, 14, 66, 70, 74, 23, 82, 9, 90, 17, 98, 102, 10, 110, 15, 25, 122, 126, 16, 39, 48, 40, 21, 150, 34, 158, 29, 54, 48, 30, 13, 182, 63, 55, 194, 56, 202, 14, 45, 214, 63, 222, 26, 41, 234, 31, 42, 39, 250, 32, 63

Offset

0,2

Comments

Triangular(k) = A000217(k) = k*(k+1)/2.

a(n) <= 4*n - 2, because with k = 4*n-2: n^2 + k*(k+1)/2 = n^2 + (4*n-2)*(4*n-1)/2 = 9*n^2 - 6*n + 1 = (3*n-1)^2.

The sequence of numbers n such that a(n)=n begins: 8, 800, 7683200 ... - a subsequence of A220186.

Links

Table of n, a(n) for n=0..65.

Mathematica

lpk[n_]:=Module[{k=1}, While[!IntegerQ[Sqrt[n^2+(k(k+1))/2]], k++]; k]; Array[ lpk, 70, 0] (* Harvey P. Dale, May 04 2018 *)

Prog

(Python)
import math
for n in range(77):
  n2 = n*n
  y=1
  for k in range(1, 10000001):
    sum = n2 + k*(k+1)//2
    r = int(math.sqrt(sum))
    if r*r == sum:
      print(str(k), end=', ')
      y=0
      break
  if y: print('-', end=', ')
(PARI) a(n) = {k = 1; while (! issquare(n^2 + k*(k+1)/2), k++); k; } \\ Michel Marcus, Nov 20 2013

Crossrefs

Cf. A000290, A000217, A038202, A055527, A220186, A232175.

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

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).

Sequence in context: A067368 A191259 A184914 * A366047 A187884 A068977

Adjacent sequences: A232173 A232174 A232175 * A232177 A232178 A232179

Keyword

nonn

Author

Alex Ratushnyak, Nov 19 2013

Status

approved