A Sierpinski gasket is three half-sized Sierpinski gaskets arranged in a triangle.

- In C it is legal for functions to call themselves. This
is known as
*recursion*. - Some computations lend themselves naturally to recursive implementations.
- Recursive functions should
- simplify the implementation of a solution
- have a
*base case* - be used sparingly

n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

F(n) | 1 | 1 | 2 | 3 | 5 | 8 | 13 | 21 | 34 | 55 |

⎧ 1 if n≤2 F(n) = ⎨ ⎩ F(n-1) + F(n-2) otherwise

fib(5) = fib(4) + fib(3) = [ fib(3) + fib(2) ] + [ fib(2) + fib(1) ] = [ fib(2) + fib(1) ] + 1 + 1 + 1 = 1 + 1 + 1 + 1 + 1 = 5

int fib(int which) { if (which <= 2) // base case? return 1; return fib(which - 1) + fib(which - 2); }

At any point, there can be several calls to `fib`

*active*.
The computer keeps track of them.

- How many ways are there to arrange:
- one card: J (1)
- two cards: JQ QJ (2×1 = 2)
- three cards: JQK JKQ QJK QKJ KJQ KQJ (3×2×1 = 6)
- four cards: … (4×3×2×1 = 24)

- In general, there are n × (n-1) × (n-2) × … × 3 × 2 × 1
ways to arrange
*n*cards. - This function is called
*factorial*: n! - 1! = 1
- n! = n × (n-1)!

int factorial(int n) { if (n <= 1) return 1; return n * factorial(n-1); }

- The function divides the input into two cases.
`n`

≤ 1 : return 1`n`

> 1 : return`n*factorial(n-1)`

- Since we are subtracting 1 from
`n`

at each successive call to factorial the recursion will stop when`n`

becomes 1.

- Take factorial(4) for example:
- factorial(4) → return 4*factorial(3)
- factorial(3) → return 3*factorial(2)
- factorial(2) → return 2*factorial(1)
- factorial(1) → return 1
- factorial(2) → return 2*1 → return 2
- factorial(3) → return 3*2 → return 6
- factorial(4) → return 4*6 → return 24

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Colorado State University, Fort Collins, CO 80523 USA

© 2018 Colorado State University