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Ordered Sums

Let a(n) be the number of ways of expressing the integer n as an ordered sum of 1's and 2's. Let b(n) be the number of ways of expressing n as an ordered sum of integers greater than 1. (i) Calculate a(n) and b(n) for n<8. What do you notice about these sequences? (ii) Find a relation between a(p) and b(q). (iii) Prove your conjectures.

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Gnomon Dimensions

Age 14 to 16 Challenge Level:

You may wish to try the related problem Building Gnomons first.

A Gnomon is a rectangle with another rectangle cut out of one corner. The area of each Gnomon is a Fibonacci number. (The Fibonacci numbers are $1, 1, 2, 3, 5, 8$ and so on, with each new term being the sum of the previous two terms.)

$G_1$ has area $3$, $G_2$ has area $5$, $G_3$ has area $8$ and so on.

Draw the next three gnomons in the sequence.

Look at the length and width of the large rectangle from which each gnomon is made.
Large gnomon
Is there a pattern to the lengths and widths?
Can you generalise?

Now look at the length and width of the rectangle cut out of each gnomon. Can you see any patterns here? Can you generalise and justify what you see?

I want to group the gnomons with area 3, 8 and 21 together, and the gnomons with area 5, 13 and 34 together. Can you explain why I want to divide these into two separate groups?

Can you give a convincing argument why all the gnomons fit into one of these two groups?

What other interesting patterns and relationships can you find?