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

1,2

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, ...

Links

Table of n, a(n) for n=1..45.

Example

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.

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 *)

Prog

(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=', ')

Crossrefs

Cf. A000217, A000290, A234141, A234142.

Sequence in context: A317378 A338866 A308485 * A217686 A042885 A042085

Adjacent sequences: A234140 A234141 A234142 * A234144 A234145 A234146

Keyword

nonn

Author

Alex Ratushnyak, Dec 19 2013

Extensions

Name corrected by Alex Ratushnyak, Jun 02 2016

Status

approved