Everyone gets a share of the pi

While reading about the history of mathematics in Japan I found a mathematical sequence which, for reasons I do not know, happens to contain a lot of the estimates for pi that have been put forward throughout history.

This sequence was discovered by Seki Kōwa and was published in 1712 after his death in a book called Katsuyō Sanpō. It runs as follows:

Start with the fraction 3/1. If this number is lower than pi, then increase the upper number by 4. If it is greater, then increase it by three. Then increase the lower number by one. And continue like this.

If you do this, you get a sequence of numbers that includes the estimates of pi from different people from different times in history. In this list (mostly Chinese mathematicians, since up until then Japanese maths was heavily influenced by Chinese textbooks), Kōwa finds the values from Chih (25/8 or 3.125), T'ung Ling (63/20, or 3.15), the “old Japanese” value (79/25 or 3.16), Liu Chi (142/45 or 3.155), Hui (157/50 or 3.14) and also 355/113 (which is very close to the actual value of pi at 3.14159292) but this isn't credited to anyone, so Kōwa must've known about it's use as a value of pi, but not known that Tsu Ch'ung-chih had found it.

I like mathematical sequences that seem to have a narrative, although I doubt it has any use. I don't think the series converges on the true value of pi (I haven't checked) but it does occur to me that this isn't a very good way of calculating pi, because it only works if you already know the value of pi.

Reference
D.E. Smith, Y. Makami, “A History of Japanese Mathematics”, Open Court Publishing Company, Chicago, 1914, p111-112