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A332590 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. 2
44, 104, 209, 90, 377, 86, 629, 285, 989, 104, 244, 1484, 322, 209, 2144, 365, 923, 144, 132, 207, 3002, 494, 4094, 1089, 5459, 363, 390, 924, 650, 7139, 7749, 714, 714, 531, 9179, 1287, 11627, 924, 6519, 1364, 1085, 755, 14534, 1517, 406, 2561, 17954, 21944, 3689 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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).

LINKS

Giovanni Resta, Table of n, a(n) for n = 1..162

Ulas Maciej, A note on Sierpinski's problem related to triangular numbers, arXiv:0810.0222 [math.NT], 2008. See Table 1.

Ulas Maciej, A note on Sierpinski's problem related to triangular numbers, Colloq. Math. 117 (2009), no. 2, 165-173. See MR2550124. See Table 1.

EXAMPLE

The initial values of x, y, z, p, q, r are:

   x    y    z    p    q    r

  --  ---  ---  --- ----  ---

   9   13   44   16   46   45

  14   51  104   53  116  105

  20   50  209   54  215  210

  23   30   90   38   95   93

  27  124  377  127  397  378

  35   65   86   74  108   93

  35  123  629  128  641  630

  41  119  285  126  309  288

  44  245  989  249 1019  990

  ...

MATHEMATICA

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 *)

CROSSREFS

Cf. A000217, A332588-A332593.

Sequence in context: A039527 A253391 A050944 * A187933 A118483 A044231

Adjacent sequences:  A332587 A332588 A332589 * A332591 A332592 A332593

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Feb 29 2020

EXTENSIONS

Terms a(10) and beyond from Giovanni Resta, Mar 02 2020

STATUS

approved

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Last modified May 14 04:27 EDT 2021. Contains 343872 sequences. (Running on oeis4.)