Trithagorean Triples A Pythagorean Triple is a positive integer solution to the equation:



A Trithagorean triple is a positive integer solution to the equation:



Where Δn finds the nth triangular number.  All Trithagorean triples are also solutions to the equation:



Task

Given a positive integer c, output all the pairs of positive integers a,b such that the sum of the ath and bth triangular numbers is the cth triangular number.  You may output the pairs in whatever way is most convenient.  You should only output each pair once.

This is code-golf

Test Cases

2: []
3: [(2, 2)]
21: [(17, 12), (20, 6)]
23: [(18, 14), (20, 11), (21, 9)]
78: [(56, 54), (62, 47), (69, 36), (75, 21), (77, 12)]
153: [(111, 105), (122, 92), (132, 77), (141, 59), (143, 54), (147, 42), (152, 17)]
496: [(377, 322), (397, 297), (405, 286), (427, 252), (458, 190), (469, 161), (472, 152), (476, 139), (484, 108), (493, 54), (495, 31)]
1081: [(783, 745), (814, 711), (836, 685), (865, 648), (931, 549), (954, 508), (979, 458), (989, 436), (998, 415), (1025, 343), (1026, 340), (1053, 244), (1066, 179), (1078, 80), (1080, 46)]
1978: [(1404, 1393), (1462, 1332), (1540, 1241), (1582, 1187), (1651, 1089), (1738, 944), (1745, 931), (1792, 837), (1826, 760), (1862, 667), (1890, 583), (1899, 553), (1917, 487), (1936, 405), (1943, 370), (1957, 287), (1969, 188)]
2628: [(1880, 1836), (1991, 1715), (2033, 1665), (2046, 1649), (2058, 1634), (2102, 1577), (2145, 1518), (2204, 1431), (2300, 1271), (2319, 1236), (2349, 1178), (2352, 1172), (2397, 1077), (2418, 1029), (2426, 1010), (2523, 735), (2547, 647), (2552, 627), (2564, 576), (2585, 473), (2597, 402), (2622, 177), (2627, 72)]
9271: [(6631, 6479), (6713, 6394), (6939, 6148), (7003, 6075), (7137, 5917), (7380, 5611), (7417, 5562), (7612, 5292), (7667, 5212), (7912, 4832), (7987, 4707), (8018, 4654), (8180, 4363), (8207, 4312), (8374, 3978), (8383, 3959), (8424, 3871), (8558, 3565), (8613, 3430), (8656, 3320), (8770, 3006), (8801, 2914), (8900, 2596), (8917, 2537), (9016, 2159), (9062, 1957), (9082, 1862), (9153, 1474), (9162, 1417), (9207, 1087), (9214, 1026), (9229, 881), (9260, 451), (9261, 430), (9265, 333)]
Mathematica, 53 49 48 bytes

Solve[{x.(x+1)==#^2+#,a&gt;=b&gt;0},x={a,b},Integers]&amp;


Example:

In[1]:= Solve[{x.(x+1)==#^2+#,a&gt;=b&gt;0},x={a,b},Integers]&amp;[21]

Out[1]= {{a -&gt; 17, b -&gt; 12}, {a -&gt; 20, b -&gt; 6}}