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enter image description here Assume we grow from a single triangle layer by layer to get the whole disk. Every time a new ring of triangles makes all the vertices of the triangles already in the picture surround by seven triangles. How many triangles are there in the nth triangle? Any hint?

Currently I get 1, 15, 180.

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I believe the official term for your layer/ring notion is "corona" in tiling lore. This may help you with web searches. (You may have to sift through numerous hits related to tiling graphic elements in the Corona app-creation program.) – Blue Sep 23 '13 at 11:14
This is a beautiful picture. I think, it is generated b reflections using (7,7,7) Coxeter group acting on the hyperbolic plane. – studiosus Sep 23 '13 at 19:27

According to this result in OEIS, the sequence you may be looking for is: $$a_n=\frac{n(n+1)(n+3)!}{48}. $$

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This is not barycenteric subdivision of anything and its rate of growth I think, is only exponential, not factorial, in n, but I have to think how to prove it. – studiosus Sep 23 '13 at 19:25

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