3. Define the sequence Rn of semi-centered rectangular numbers as depicted here:

Question

3. Define the sequence Rn of semi-centered rectangular numbers as depicted here: R3-15 R4-28 Each rectangular array of dots is formed by adding one layer of dots to the left side, right side, and top of the previous rectangular array a) 5 Find an equation that relates Rn to the triangular numbers. Show your equation on this (or your own) diagram: b) [5 Find an equation that relates Rn to the square numbers. Show your equation on this (or your own) diagram c) 5 Find a recurrence relation involving Rn and R1

Explanation / Answer

Consider the 7 pigeonholes given by the sets {1,2,4,8},{3,6,12},{5,10},{7,14},{9},{11},{13}

These are the sets containing distinct odd numbers and their multiples by a power of 2 that are not larger than 14.

When the 8 numbers are placed into 7 pigeonholes, two of the numbers belong to the same set.

Since the smaller number divides the larger,we are done

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