module Laziness where

import Prelude
import CFrac
import Ratio

depth :: Integer -> Integer -> Integer
-- depth p q is the depth of the regular tunnel 
-- of the (p,q) torus knot
depth pp qq 
  | pp == 0 && qq == 0    = 0
  | abs pp < 3            = 0 
  | abs qq == 1           = 0 
  | gcd pp qq  /= 1       = 0 
  | otherwise             = count ( tail expansion)
  where
  expansion  = cFrac (abs pp % abs qq)
  count [ ] = 0
  count [n] = 1
  count (1:n:ns) = 1 + (count ns)
  count (m:n:ns) = 1 + (count (n:ns))

depths :: Integer -> [Integer]
-- a list of the depths of the (p,*) torus knot tunnels, p > q
depths p = map (depth p) [2..p-1]

firstDepthN :: Integer -> Integer
-- the first p for which there is a tunnel of depth n for
-- some (p,q) torus knot
firstDepthN n = head [ p | p <- [2..], maximum ( depths p ) >= n ]

{-
Sample output:

Laziness> firstDepthN 1
3
Laziness> firstDepthN 2
7
Laziness> firstDepthN 3
17
Laziness> firstDepthN 4
41
Laziness> firstDepthN 5
99
Laziness> firstDepthN 6
239
Laziness> firstDepthN 7
577

-}