Date: Sat, 14 Mar 2009 21:46:16 -0400
Subject: A069693 .. A069700 (triangular numbers of the form abb...bc) are fini and full
From: Max Alekseyev (maxale(AT)gmail.com)

Each of the sequences A069693 .. A069700 is finite and complete.

Here is the proof:

Let a and c be external digits and b be an internal digit of a
triangular number:
a bb...bb c  = a*10^(n-1) + (10^(n-2)-1)/9 * b * 10 + c
where n is the length of this triangular number.
We can also represent this triangular number in the form (m^2-1)/8,
implying the equation:
a*10^(n-1) + (10^(n-2)-1)/9 * b * 10 + c = (m^2 - 1)/8
or
(72a + 8b)*10^(n-1) - 9m^2 + (72c - 80b + 9) = 0.
Integer solutions to this equation correspond to (some) integral
points on the following three Mordell curves indexed by k = 0, 1, or
2:
(72a + 8b) * 10^k * x^3 - 9y^2 + (72c - 80b + 9) = 0.
Here k stands for a possible value of (n-1) mod 3.
It is well known that the number of integral points is finite (for
every possible choice of digits a,b,c).

Solutions to these equations (which can be obtained in MAGMA or SAGE)
lead to following finite set of triangular numbers have the required form:
0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136,
153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465,
496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946, 990,
1225, 1770, 2556, 2775, 3003, 4005, 5778, 5886, 5995, 6441, 6555,
8001, 8778, 21115, 46665, 333336, 544446, 5666661
Sequences A069693 .. A069700 represent subsets of this set.