Discrete Mathematics- An Open Introduction - 3rd Edition, 2016a

2.3. Polynomial Fitting 161
sequence ∆ 3 -constant. The sequence 1, 4, 9, 16, . . . has second differences
constant, so it will be a ∆ 2 -constant sequence. In general, we will say a
sequence is a ∆ k -constant sequence if the kth differences are constant.
Example 2.3.1
Which of the following sequences are ∆ k -constant for some value of
k?
1. 2, 3, 7, 14, 24, 37, . . ..
2. 1, 8, 27, 64, 125, 216, . . ..
3. 1, 2, 4, 8, 16, 32, 64, . . ..
Solution.
1. This is the sequence from Example 2.2.6, in which we found a
closed formula by recognizing the sequence as the sequence of
partial sums of an arithmetic sequence. Indeed, the sequence
of first differences is 1, 4, 7, 10, 13, . . ., which itself has differences
3, 3, 3, 3, . . .. Thus 2, 3, 7, 14, 24, 37, . . . is a ∆ 2 -constant
sequence.
2. These are the perfect cubes. The sequence of first differences
is 7, 19, 37, 61, 91, . . .; the sequence of second differences is
12, 18, 24, 30, . . .; the sequence of third differences is constant:
6, 6, 6, . . .. Thus the perfect cubes are a ∆ 3 -constant sequence.
3. If we take first differences we get 1, 2, 4, 8, 16, . . .. Wait, what?
That’s the sequence we started with. So taking second differences
will give us the same sequence again. No matter
how many times we repeat this we will always have the same
sequence, which in particular means no finite number of differences
will be constant. Thus this sequence is not ∆ k -constant
for any k.
The ∆ 0 -constant sequences are themselves constant, so a closed formula
for them is easy to compute (it’s just the constant). The ∆ 1 -constant
sequences are arithmetic and we have a method for finding closed formulas
for them as well. Every ∆ 2 -constant sequence is the sum of an arithmetic
sequence so we can find formulas for these as well. But notice that the
format of the closed formula for a ∆ 2 -constant sequence is always quadratic.
For example, the square numbers are ∆ 2 -constant with closed formula
a n n 2 . The triangular numbers (also ∆ 2 -constant) have closed formula
a n n(n+1)
2
, which when multiplied out gives you an n 2 term as well. It