A very simple way of generating a sequence is to take two starting values, a1 and a2, and construct a3 as their arithmetic mean, a4 as the mean of a2 and a3 and so on. This sequence always converges to the limit and seems to be fairly widely known as a topic for investigation in schools and colleges. It can immediately be generalised to the case of k starting values, and this forms a good subject for investigation using a micro-computer. It seems likely that such a simple set-up has arisen independently many times. One emergence occurred about ten years ago at a conference of the Association of Teachers of Mathematics, where the originator introduced it under the fictitious name ‘Littov sequences’. It has recently resurfaced at a Department of Education and Science conference.