# "simple" series not in OEIS yet?

Peter Pein petsie at dordos.net
Wed Jun 13 07:35:13 CEST 2007

```Dear seqfans,

when looking for the triangular numbers, which are differences of nonnegative
cubes, I can't find them. Using Mathematica, I got

Reduce[{i^3 - j^3 ==  k*((k + 1)/2) == m >= 0, i > j >=0 <= k}, {i, j, k, m},
Integers, Backsubstitution -> True]
-->
(i == 1 && j == 0 && k == 1 && m == 1) ||
((i | j | k) \[Element] Integers && i >= 2 && 0 <= j <= -1 + i &&
k == (1/2)*(-1 + Sqrt[1 + 8*i^3 - 8*j^3]) &&
m == i^3 - j^3)

which leads to the following algorithm (to be sure, there are no gaps, I
select the solutions which are less than n^3-(n-1)^3) - where n is the maximal
value of the "i" above:

an = With[{n = 1000},
Select[Union[Apply[ #1^3 - (#2 - 1)^3 & ,
Position[Outer[(1/2)*(-1 + Sqrt[1 + 8*#1^3 - 8*#2^3]) & ,
Range[n], Range[0, n - 1]], x_Integer /; x >= 0, {2}], {1}]],
1 - 3*n + 3*n^2 >= #1 & ]]

--> {0, 1, 91, 4095, 5886, 7875, 8128, 8911, 9045, 17955, 21736, 23653, 47278,
93961, 115921, 130816, 184528, 259560, 379756, 488566, 575128, 658378,
758296, 810901, 873181, 885115, 1060696, 1155960, 1358776, 1385280, 1997001,
2616328, 2685403}

Is there a mistake (in my thoughts, or in Mathematica)? And might there be an
interest in this sequence? At least, it is not base-dependent.

I sent the elements starting with 1 and ending with 115921 to the superseeker
program six hours ago, but did not receive any answer until now.

Best regards,
Peter

```

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