How do you keep an iterable alive to keep pulling from it?

I was looking at the https://docs.flutter.dev/get-started/codelab tutorial for dart, and I noticed that they kept recalling their iterator. I wanted to keep the iterator around, and just continue to pull from it, so I worked up this:

class _FunPageState extends State<FunPage> {
  final _things = <Widget>[];
  late Iterable<Widget> _funPuller;
  @override
  void initState() {
    super.initState();
    _funPuller = generateFunThings();
  }
  Iterable<Widget> generateFunThings() sync*{
    int i=1;
    while (true) {
      yield Text("$i");
      i += 1;
    }
  }
  @override
  Widget build(BuildContext context) => ListView.builder(
    padding: const EdgeInsets.all(8),
    itemBuilder: (context, index) {
      if (index.isOdd) {
        return const Divider();
      }
      index = (index/2).floor();
      if (index >= _things.length) {
        _things.addAll(_funPuller.take(2));
      }
      return _things[index];
    },
  );
}

Except I noticed that this just generated the list 1,2,1,2,1,2,... rather than 1,2,3,4,5,... as I expected. Is there a way I can get the result of generateFunThings() to persist so I can continue calling it?