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A308395 Numbers y such that x*(x+1) + y*(y+1) = z*(z+1) is solvable in positive integers x, z with x <= y.

%I A308395 #96 Aug 02 2019 18:56:27
%S A308395 2,5,6,9,10,13,14,17,18,20,21,22,24,25,26,27,29,30,33,34,35,37,38,39,
%T A308395 41,42,44,45,46,48,49,50,51,53,54,55,56,57,58,61,62,65,66,68,69,70,73,
%U A308395 74,75,76,77,78,80,81,82,84,85,86,89,90,91,92,93,94,95,97
%N A308395 Numbers y such that x*(x+1) + y*(y+1) = z*(z+1) is solvable in positive integers x, z with x <= y.
%H A308395 Chai Wah Wu, <a href="/A308395/b308395.txt">Table of n, a(n) for n = 1..10000</a>
%e A308395 14 is a term because 14*15 + 14*15 = 20*21.
%t A308395 max = 220; lst = {}; For[x = 1, x < max, x++,
%t A308395 For[y = x, y < max, y++,
%t A308395   For[z = y, z < max, z++,
%t A308395    If[x (x + 1) + y (y + 1) == z (z + 1),
%t A308395     lst = AppendTo[lst, y]]]]]; Select[Union[lst], # < max/2 &]
%o A308395 (Python)
%o A308395 from sympy import integer_nthroot
%o A308395 A308395_list, y, w = [], 1, 0
%o A308395 while len(A308395_list) < 10000:
%o A308395     w += y
%o A308395     z = 0
%o A308395     for x in range(1,y+1):
%o A308395         z += x
%o A308395         if integer_nthroot(8*(w+z)+1,2)[1]:
%o A308395             A308395_list.append(y)
%o A308395             break
%o A308395     y += 1 # _Chai Wah Wu_, Aug 02 2019
%Y A308395 Cf. A012132.
%K A308395 nonn
%O A308395 1,1
%A A308395 _Ralf Steiner_, Jul 31 2019