My Math Forum cubed number sequence

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 April 28th, 2015, 02:07 AM #1 Newbie   Joined: Apr 2015 From: uk Posts: 1 Thanks: 0 cubed number sequence the sequence of cubed numbers increases by a sequence of numbers that differ by the six times table ie 0,1,8,27,64 1,7,19,37 (the difference) 6,12,18,24 (the second row of difference) my question is why is that?!
 April 28th, 2015, 03:04 AM #2 Global Moderator   Joined: Dec 2006 Posts: 20,919 Thanks: 2203 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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