fibonacci series what Iam is missing ?

[CHIN Dihedral]

On Tuesday, March 24, 2015 at 6:54:20 AM UTC+8, Terry Reedy wrote:
> On 3/23/2015 2:44 PM, Dave Angel wrote:
> 
> > ## Example 2: Using recursion with caching
> > cache = [0, 1]
> > def fib4(n):
> >      if len(cache) <= n:
> >          value = fib4(n-2) + fib4(n-1)
> >          cache.append(value)
> >      return cache[n]
> >
> > This one takes less than a millisecond up to n=200 or so.
> 
> Iteration with caching, using a mutable default arg to keep the cache 
> private and the function self-contained.  This should be faster.
> 
> def fib(n, _cache=[0,1]):
>      '''Return fibonacci(n).
> 
>      _cache is initialized with base values and augmented as needed.
>      '''
>      for k in range(len(_cache), n+1):
>          _cache.append(_cache[k-2] + _cache[k-1])
>      return _cache[n]
> 
> print(fib(1), fib(3), fib(6), fib(5))
> # 1 2 8 5
> 
> Either way, the pattern works with any matched pair of base value list 
> and recurrence relation where f(n), n a count, depends on one or more 
> f(k), k < n.  'Matched' means that the base value list is as least as 
> long as the maximum value of n - k.  For fib, the length and max are both 2.
> 
> -- 
> Terry Jan Reedy
How about adding a new type of 
numbers of the form:
 (a+sqrt(r0)+ swrt(r1)....)/b with proper operations, and  then one can compute the Fib term directly in a 
field.