Thread: sequence formula

I need find a closed form for this recurrence relation:

b sub n = b sub n-1 + n^2 for n >0 and b sub 0 = 3

Anybody know how to do this?

Originally Posted by jzellt

I need find a closed form for this recurrence relation:

b sub n = b sub n-1 + n^2 for n >0 and b sub 0 = 3

Anybody know how to do this?

There is probably a more elegant way than this:

1. Calculate the first five values
2. Calculate the differences between consecutive values
3. Repeat this process until you get a constant sequence.
4. Determine the number of repetitions. That is the degree of the closed form:

Code:

degree 3:  3    4    8     17    33    ...
degree 2:     1    4    9     16    ...
degree 1:       3    5     7    ...
degree 0:         2     2

That means you are looking for a term with degree 3:



From the initial row you know:
s_0 = 3
s_1 = 4
s_2 = 8
s_3 = 17

Thus you have to solve a system of simultaneous equations:


After a few steps you'll get: