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%I #19 May 05 2021 13:39:48
%S 1,4,5,19,22,25,40,64,85,89,110,124,127,148,263,552,688,700,705,790,
%T 1804,2101,4009,4108,8680,11830,15889,22125,23611,23710,27571,32902,
%U 34536,39520,47327,62329,68374,98896,100933,112660,137614,137989,138191,159124,205004
%N 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).
%C Intersection of A234141 and A234142.
%C The sequence of triangular(a(n)) begins: 1, 10, 15, 190, 253, 325, 820, 2080, 3655, 4005, 6105, 7750, 8128, ...
%e 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.
%t 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 *)
%o (Python)
%o import math
%o def isTriangular(n): # OK for relatively small n
%o n+=n
%o sr = int(math.sqrt(n))
%o return (n==sr*(sr+1))
%o for n in range(1,264444):
%o tn = n*(n+1)//2
%o r = int(math.sqrt(tn-1))
%o i = tn-r*r
%o r = int(math.sqrt(tn))
%o j = (r+1)*(r+1)-tn
%o if isTriangular(i) and isTriangular(j): print(str(n), end=',')
%Y Cf. A000217, A000290, A234141, A234142.
%K nonn
%O 1,2
%A _Alex Ratushnyak_, Dec 19 2013
%E Name corrected by _Alex Ratushnyak_, Jun 02 2016