|Prime spiral (Ulam spiral)|
A difference table is made by listing the terms of a sequence and its differences. It includes the first differences, which is a sequence that lists the differences of two consecutive terms of the original sequence. For instance, the first term of the first differences is the difference between the first and second term of the original sentence, the second term of the first difference is the difference between the second and third term of the original sentence, and so forth.
Second-order differences, which is a sequence of differences of the first differences, and other higher-order differences can also be included in the difference table.
An important thing to notice is that we find the differences by subtracting the earlier term from the later term of the sequence and not by subtracting the term with the smaller value from the term with the larger value. Thus, it is possible to have a negative number in the difference table.
For example, let's create a difference table for the sequence of perfect squares :
First, list the terms of the sequence on the top row. Then write each difference of two consecutive terms underneath and in between the terms it is the difference of. For instance, the terms in the first differences are found by . We can also create another row of second-order differences that lists the differences of two consecutive terms from the first sequence of differences.
As we can see from this difference table, the last row of third-degree differences consists of 's that continue infinitely. Indeed, all polynomial sequences generate a row of only 's at some level of the difference table. Every sequence that eventually reduces to 's can be written as a polynomial sequence. Furthermore, there are some methods that we can use to determine the specific polynomial function.
The table below shows a difference table of the Fibonacci sequence.. Because Fibonacci numbers are not polynomials, they do not reach a row with only 's.
For more information about the Fibonacci numbers and the difference table of Fibonacci numbers, go to Finite Difference of Fibonacci Numbers.