The pyramid's formula

Building a whole pyramid just to find the top number is slow. Algebra gives a shortcut.

Take a three-row pyramid with bottom row a, b, c. The middle row is a + b and b + c. The top is their sum: (a + b) + (b + c) = a + 2b + c.

So the top is not just the sum of the bottom row. The middle number counts twice, because it feeds into both boxes above it.

Now the top of any three-row pyramid takes one line. Bottom row 8, 13, 4? Top = 8 + 2(13) + 4 = 38. No pyramid needed.

Four rows, with bottom row a, b, c, d? Work it through the same way and you get a + 3b + 3c + d.

The numbers in front are 1, 2, 1 for three rows and 1, 3, 3, 1 for four. You may know this pattern from Pascal's triangle. Each number tells you how many different paths lead from that bottom box up to the top.

Now a lovely question. Remember the Virahāṅka-Fibonacci sequence 1, 2, 3, 5, 8, 13, …, where each number is the sum of the two before it.

Put its first three numbers, 1, 2, 3, in the bottom row of a three-row pyramid. The middle row is 1 + 2 = 3 and 2 + 3 = 5. The top is 3 + 5 = 8. Every number in the pyramid is a Virahāṅka-Fibonacci number.

That is no accident. The pyramid's rule (add the two below) is the same rule as the sequence's. So it works for four rows, for 29 rows, and for an n-row pyramid built on the first n Virahāṅka-Fibonacci numbers: the whole pyramid stays inside the sequence, and the top is a Virahāṅka-Fibonacci number too.