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%I #16 Mar 02 2020 20:08:53
%S 44,104,209,90,377,86,629,285,989,104,244,1484,322,209,2144,365,923,
%T 144,132,207,3002,494,4094,1089,5459,363,390,924,650,7139,7749,714,
%U 714,531,9179,1287,11627,924,6519,1364,1085,755,14534,1517,406,2561,17954,21944,3689
%N Let t_k denote the triangular number k*(k+1)/2. Suppose 0 < x < y < z are integers satisfying t_x + t_y = t_p, t_y + t_z = t_q, t_x + t_z = t_r, for integers p,q,r. Sort the triples [x,y,z] first by x, then by y. Sequence gives the values of z.
%C Ulas gives a table assuming 0 < x < y < z < 1000. Because of the assumption z < 1000, only the entries with x < 46 can be relied upon (above this it is possible that there are gaps in the table).
%H Giovanni Resta, <a href="/A332590/b332590.txt">Table of n, a(n) for n = 1..162</a>
%H Ulas Maciej, <a href="https://arxiv.org/abs/0810.0222">A note on Sierpinski's problem related to triangular numbers</a>, arXiv:0810.0222 [math.NT], 2008. See Table 1.
%H Ulas Maciej, <a href="http://dx.doi.org/10.4064/cm117-2-2">A note on Sierpinski's problem related to triangular numbers</a>, Colloq. Math. 117 (2009), no. 2, 165-173. See MR2550124. See Table 1.
%e The initial values of x, y, z, p, q, r are:
%e x y z p q r
%e -- --- --- --- ---- ---
%e 9 13 44 16 46 45
%e 14 51 104 53 116 105
%e 20 50 209 54 215 210
%e 23 30 90 38 95 93
%e 27 124 377 127 397 378
%e 35 65 86 74 108 93
%e 35 123 629 128 641 630
%e 41 119 285 126 309 288
%e 44 245 989 249 1019 990
%e ...
%t L = {}; t[n_] := n (n + 1)/2; Do[ syp = Solve[t[x] + t[y] == t[p] && p > 0 && y > x , {p, y}, Integers]; If[syp =!= {}, Do[{y1, p1} = {y, p} /. e; s = Solve[ t[y1] + t[z] == t[q] && t[x] + t[z] == t[r] && q > 0 && z > y1 && r > 0, {z, q, r}, Integers]; If[s =!= {}, L = Join[L, {x, y1, z, p1, q, r} /. s]], {e, syp}]], {x, 54}]; Sort[L][[All, 3]] (* _Giovanni Resta_, Mar 02 2020 *)
%Y Cf. A000217, A332588-A332593.
%K nonn
%O 1,1
%A _N. J. A. Sloane_, Feb 29 2020
%E Terms a(10) and beyond from _Giovanni Resta_, Mar 02 2020
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