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April 28th, 2015, 02:07 AM   #1
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cubed number sequence

the sequence of cubed numbers increases by a sequence of numbers that differ by the six times table
1,7,19,37 (the difference)
6,12,18,24 (the second row of difference)

my question is why is that?!
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April 28th, 2015, 03:04 AM   #2
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As the original numbers are cubes, the first differences are given by a quadratic expression and the second differences by a linear expression. It happens to be the case that this linear expression has no constant term, so the second differences form a times table.

Had you started with cube plus square, you would have got
0, 2, 12, 36, 80, . . .
2, 10, 24, 44, . . . (first differences)
8, 14, 20, . . . (second differences)
so the second differences no longer form a times table.

Note that if you start with nth powers, the nth differences all equal n factorial.
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