[seqfan] A069693 .. A069700 (triangular numbers of the form abb...bc) are fini and full
Max Alekseyev
maxale at gmail.com
Sun Mar 15 02:46:16 CET 2009
Neil,
Each of the sequences A069693 .. A069700 is finite and complete.
Can you please indicate this fact in the sequences?
Here is the proof outline:
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.
Other minor corrections:
1) it makes sense to prepend 0 to each of these sequence;
2) there is also a typo in all these sequences:
instead of
%Y A069700 Cf. A069675 to A069684, A069693 to A067000.
it should be
%Y A069700 Cf. A069675 to A069684, A069693 to A069700.
Regards,
Max
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