Unnecessary Triangles

[Ed Pegg]

 From my mathpuzzle.com page.

  I pondered a possible game this weekend, where cards or dominoes would
represent the first 14 triangular numbers: 0, 1, 3, 6, 10, 15, 21, 28, 36,
45, 55, 66, 78, and 91 (D0 = 0). By summing three of these numbers, all but
one target number from 1 to 172 can be represented (which one is missing?).
I'd use 1-100 in a game. Gauss found a difficult proof that any integer was
the sum of three triangulars. I wondered if I might spot a pattern in it all
that would lead to an easier proof, but didn't. John Conway gave me some
advice: "You might like to look in my little book, The Sensual Form, which
discusses Legendre's 3-squares theorem (of which the three triangles theorem
is the particular case 8n+3) in a way that will probably help you to
understand what's going on. The proof hasn't really changed since the days
of Gauss and Legendre, and my guess is that it will remain hard."

While trying to figure out a fair distribution of cards, I noticed that
(tri)7 (28) was essential. 29, 44, 50, 53, 74, 83, 119, and 239 all require
(tri)7 for the tri+tri+tri sum. The first seven triangulars, and (tri)10 and
(tri)12, are all essential. 36 ((tri)8) seemed to be nonessential, and I
recursively constructed a whole list of nonessential (tri)s. The list starts
{8, 11, 16, 17, 23, 24, 29, 31, 36, 38, 41, 43, 45, 49, 50, 59, 60, 61, 62,
65}.

8, 11, 16, 17, 23, 24, 29, 31, 36, 38, 41, 43, 45, 49, 50, 59, 60, 61, 62,
65

It looks like a nice sequence, and I show it works for the first thousand
integers
at http://www.mathpuzzle.com/tritritri.htm ... but i'm not sure how to prove
unnecessaryness in general.   Strikes me as a hard problem.  I've ordered
Conway's little book, but haven't added the sequence yet.

Every positive integer is the sum of four squares, and Richard Schroeppel
noted that you don't need 49 (or a bunch of others).  This would also be an
interesting sequence