A332589 - OEIS

A332589

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 y.

%I #18 Mar 02 2020 20:08:37

%S 13,51,50,30,124,65,123,119,245,69,143,244,99,135,426,195,459,90,125,

%T 125,425,284,679,678,678,170,363,551,429,1016,1547,174,245,154,1015,

%U 602,1449,260,1859,398,579,469,1448,1007,305,1275,1990,1989,1044,1575,390,429

%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 y.

%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="/A332589/b332589.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, 2]] (* _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