%I #17 Mar 03 2020 10:27:37
%S 46,116,215,95,397,108,641,309,1019,125,283,1504,337,249,2186,414,
%T 1031,170,182,242,3032,570,4150,1283,5501,401,533,1076,779,7211,7902,
%U 735,755,553,9235,1421,11717,960,6779,1421,1230,889,14606,1821,508,2861,18064,22034
%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 q.
%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="/A332592/b332592.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, 5]] (* _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