A234143 Numbers k such that triangular(k) - x and y - triangular(k) are both triangular numbers (A000217), where x is the nearest square below triangular(k), y is the nearest square above triangular(k).

1, 4, 5, 19, 22, 25, 40, 64, 85, 89, 110, 124, 127, 148, 263, 552, 688, 700, 705, 790, 1804, 2101, 4009, 4108, 8680, 11830, 15889, 22125, 23611, 23710, 27571, 32902, 34536, 39520, 47327, 62329, 68374, 98896, 100933, 112660, 137614, 137989, 138191, 159124, 205004

Offset: 1

Author

Alex Ratushnyak, Dec 19 2013

Keywords

• nonn

Comments

Intersection of A234141 and A234142.

The sequence of triangular(a(n)) begins: 1, 10, 15, 190, 253, 325, 820, 2080, 3655, 4005, 6105, 7750, 8128, ...

Examples

			Triangular(4) = 4*5/2 = 10. The nearest squares above and below 10 are 9 and 16. Because both 10-9=1 and 16-10=6 are triangular numbers, 4 is in the sequence.
		

Crossrefs

Cf. A000217, A000290, A234141, A234142.

Programs

Mathematica

btnQ[n_]:=Module[{tr=(n(n+1))/2,x,y},x=Floor[Sqrt[tr]]^2;y=Ceiling[ Sqrt[ tr]]^2;!IntegerQ[Sqrt[tr]]&&AllTrue[{Sqrt[1+8(tr-x)],Sqrt[1+ 8(y-tr)]}, OddQ]]; Join[{1},Select[Range[205100],btnQ]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 12 2020 *)

Python

import math
def isTriangular(n):  # OK for relatively small n
  n+=n
  sr = int(math.sqrt(n))
  return (n==sr*(sr+1))
for n in range(1,264444):
  tn = n*(n+1)//2
  r = int(math.sqrt(tn-1))
  i = tn-r*r
  r = int(math.sqrt(tn))
  j = (r+1)*(r+1)-tn
  if isTriangular(i) and isTriangular(j):  print(str(n), end=',')

Extensions

Name corrected by Alex Ratushnyak, Jun 02 2016