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Start with a square. Then another, the same size.
1, then 1. Two tiles. That is the whole beginning.
No spiral yet. Just two equal floors, side by side. Each new square in this picture is as big as the last two put together. So the next one has to be 2. Then 3. Then 5. The rule is almost rude: add the last two. Keep going.
People later named the list. You can see it without the name: 1, 1, 2, 3, 5, 8…