# Fibonacci numbers (Sed)

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The Fibonacci numbers are the integer sequence 0, 1, 1, 2, 3, 5, 8, 13, 21, ..., in which each item is formed by adding the previous two. The sequence can be defined recursively by

- 1 \\ \end{cases} ."/>

Fibonacci number programs that implement this definition directly are often used as introductory examples of recursion. However, many other algorithms for calculating (or making use of) Fibonacci numbers also exist.

Although `sed`

is capable of decimal arithmetic when pushed, the direct approach is to restrict ourselves to unary arithmetic, allowing straight transcription of the definition.

The first two cases might be transcribed as:

s/// s/1/1/

but as they are identities, we omit them, leaving the final case, which we first rewrite into the form

*F*(*n*+ 2) =*F*(*n*+ 1) +*F*(*n*)

and then transcribe as:

<<fibonacci.sed>>=:F s/11\(1*\)/1\1+\1/ t F

The code above repeats until convergence, at which point F has been fully applied, but the resulting additions have yet to be evaluated. In unary, this is also straightforward: addition is concatenation.

<<fibonacci.sed>>=s/+//g

## testing

To test, we sugar the output a little

<<fibonacci.sed>>=s/^/ = /

and ask for a few fibonacci numbers:

> sed -f fibonacci.sed 1 = 1 11 = 1 111 = 11 1111 = 111 11111 = 11111 111111 = 11111111 1111111 = 1111111111111 11111111 = 111111111111111111111

Unary rapidly becomes annoying for larger numbers, but a simple wrapper can provide decimal input and output.

> jot -b 1 -s "" 16 | sed -f fibonacci.sed | tr -Cd 1 | wc -c 987 > jot -b 1 -s "" 20 | sed -f fibonacci.sed | tr -Cd 1 | wc -c 6765

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