Read the sections on Fibonacci numbers in Chapter 3. We will skip Catalan numbers until later. Review the sections on triangular and tetrahedral numbers to see the connection with Pascal's triangle.

0. Redo the last problem from Assignment #2 using Newton's method. The first problem below is an attempt to explain this method.

Also check out some of the Web Pascal triangle calculators and references that show the fractal patterns.

The figurate numbers are sequences of numbers such as the triangular numbers. These are number that some from the diagonal of Pascal's triangle that is parallel to the left side, so we have

• F0(n) is 1, 1, 1, 1, … for all n = 0, 1, 2, 3, …
• F1(n) is 1, 2, 3, 4, … for all n = 0, 1, 2, 3, … (segment numbers)
• F2(n) is 1, 3, 6, 10, … for all n = 0, 1, 2, 3, … (triangular numbers)
• F3(n) is 1, 4, 10, 20, … for all n = 0, 1, 2, 3, … (tetrahedral numbers)

Etc.

You can think of these as functions on the positive integers and graph them either with bar graphs or with points in the x, y plane.

The Difference Operator D is an operator that takes such a function and produces another function. If G(n) is a function, then (DG)(n) = G(n+1) - G(n).

Think about what this means and answer the following questions.

• If G(n) = 2n + 3, what is DG(n)? (Give a formula and/or draw a graph and explain.)

• If G and H are any functions, explain why D(G+H)(n) = DG(n) + DH(n).

• If H(n) = n ², what is DH(n)?

• What is DF1(n)? What is DF2(n)? What is DF3(n)? What is the pattern?

Compute the Fibonacci numbers for many places and at each stage. Compute the ratio of the new Fibonacci divided by the previous one. Do you see a pattern? You may want to use a spreadsheet for this.

Make a series of bar graphs, each showing a row of the numbers in the Pascal triangle. But make one change. In each case divide the coefficients by the sum of all the coefficients in the row, so the numbers you plot are numbers between 0 and 1 representing the "percentage" of the total taken up by the single coefficient. You may want to use a spreadsheet to help with this, or a calculator.

Follow up one of the online sources and write up a page of some interesting aspect of the Pascal triangle and/or Fibonacci numbers.

Look up some biographical about Pascal or Fibonacci or about the Indian or Chinese history of the Pascal triangle and write a half-page about what you found.