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Triangular numbers that are the sum of two distinct nonzero triangular numbers in exactly two ways.
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%I #16 Feb 06 2022 21:29:39

%S 231,1081,1225,1431,1711,2556,5356,7381,7875,8256,15931,19306,20706,

%T 26106,30381,33153,36856,46056,51681,75078,78606,102831,104653,135981,

%U 149331,153181,182106,197506,225456,263175,265356,316410,317206,336610,379756,424581

%N Triangular numbers that are the sum of two distinct nonzero triangular numbers in exactly two ways.

%e 231 = 21 + 210 = 78 + 153.

%e 1081 = 91 + 990 = 378 + 703.

%t (P=PolygonalNumber)[3,Select[Range@200,Length@Select[Subsets[P[3,Range[s=#]],{2}],Total@#==P[3,s]&]==2&]] (* _Giorgos Kalogeropoulos_, Dec 31 2021 *)

%o (Python)

%o from collections import Counter

%o from itertools import count, takewhile, combinations as combs

%o def aupto(limit):

%o tris = takewhile(lambda x: x <= limit, (k*(k+1)//2 for k in count(1)))

%o trilst = list(tris); triset = set(trilst)

%o tri2ct = Counter(sum(c) for c in combs(trilst, 2) if sum(c) in triset)

%o return sorted(t for t in tri2ct if t <= limit and tri2ct[t] == 2)

%o print(aupto(500000)) # _Michael S. Branicky_, Dec 27 2021

%Y Intersection of A000217 and A265134.

%K nonn

%O 1,1

%A _Shyam Sunder Gupta_, Dec 27 2021