package.scala

    def convergentFractions:Iterator[(BigInt, BigInt)] = {
      val fractions = continuedFractions.memoizedByIndex
      val forDenominator = (i:Int) => fractions(i+1)
      val cache = mutable.HashMap.empty[(Int,Boolean), BigInt]
      def eval(n:Int, isNumerator:Boolean): BigInt = {
        cache.getOrElseUpdate((n, isNumerator), {
          val on = if (isNumerator) fractions else forDenominator
          n match {
            case larger if larger > 2 =>
              val a = on(larger - 2)
              val b = eval(larger - 1, isNumerator)
              val c = eval(larger - 2, isNumerator)
              a *  b + c
            case 2 => on(0)
            case 1 => BigInt(1)
            case _ => throw new IllegalArgumentException(n.toString)
          }
        })
      }
      Iterator.from(1).map { n =>
        val num = eval(n + 1, isNumerator = true)
        val denom = eval(n, isNumerator = false)
        (num, denom)
      }
    }