Introducing the Fibonacci Sequence

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Extending the Fibonacci Family

There are two easy ways to develop a new Fibonacci series:

  1. Choose different starting numbers
  2. Include more generations

It is more interesting to start with numbers that are not part of the plain vanilla Fibonacci sequence. Starting with “4, 7” continues with “11, 18, 29,47, 76, 123” and so on.

Surprisingly, any Fibonacci sequence starting with “zero, N” gives a sequence where each number is exactly N times larger than the corresponding “plain vanilla” Fibonacci number.

The proof is as follows. We already have the plain vanilla Fibonacci series x = x[n] + x when x[0] = zero and x[1] = 1. Let x[1] = N for any positive integer. Then x[2] = N+0 = N rather than just 1; x[3] = N+N rather than 2. So the new sequence starts as “0, N, N, 2N”. Divide by N, and we have the “plain vanilla” start of “0, 1, 1, 2”.

By the induction principle, we assume the new Y = Nx, and that the new Y[n] = Nx[n]. (This was true for n=1, 2 and 3). It remains to see whether Y = Nx.

Nx = N*(x[n] + x) = Nx[n] + Nx for the “plain vanilla” Fibonacci sequence, by the distribution of multiplication over addition. But we assumed that the new Y = Nx, and that the new Y[n] = Nx[n], so “Nx[n] + Nx” = “Y[n] + Y”.

Therefore Nx = Y[n] + Y, which is exactly how to calculate Y. So Y = Nx.

The other way to create a different Fibonacci series is to include more numbers. A triple series would use T = T[n] + T + T, for example. Of course, the three starting numbers could take any desired values.

Spreadsheet Exercises for the Fibonacci Series

It is easy to build a spreadsheet like the one shown below. Any cell with a non-white background colour and a number has a formula. Only the “0” and “1” in column A, and the “1” in column D, were entered manually.

“Fibonacci Series and Golden Ratio” by Mike DeHaan

After building this spreadsheet, it is easy to extend the “plain vanilla” Fibonacci sequence by copying more cells down column A. Or one might change the first two values to create a new Fibonacci series.

References:
Knott and Quinney, Plus, “The life and numbers of Fibonacci”, published Sept. 1, 1997, referenced May 22, 2011.
Scott Hotton, Mission College, “Natural Fibonacci”, copyright 1999, referenced May 23, 2011.

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