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When I see a MapServer site I usually get deeper and check how it was done. Recently I found on one server where is avalanche map strange tms url. I haven't seen it before.

Can you tell me what is this type?

I know that 14 is zoom and rest is unknown for me.

Before visiting this link you have to visit http://mapy.hiking.sk/?zoom=14&lat=6305837.94292&lon=2194989.75612&layers=00BFTTTTTTFFFFFTT to get cookies otherwise, this url below will not load.

https://mapy.hiking.sk/layers/laviny/14/000/009/088/000/010/772.png

enter image description here

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Looking at the JS script creating the map, we can see that the layer is of type OpenLayers.Layer.TileCache. This tile cache software uses a 7 components URL, with 1 for the zoom, 3 for X, and 3 for Y. Each of the 3 X and Y components contains 3 numbers at most, so the tile X = 123 456 789, Y=1, Z=10 will be shown as /layername/10/123/456/789/000/000/1.png

I haven't found an official textual definition of this, but it corresponds to the source code and to several comments.

components = ( self.basedir,
               tile.layer.name,
               "%02d" % tile.z,
               "%03d" % int(tile.x / 1000000),
               "%03d" % (int(tile.x / 1000) % 1000),
               "%03d" % (int(tile.x) % 1000),
               "%03d" % int(tile.y / 1000000),
               "%03d" % (int(tile.y / 1000) % 1000),
               "%03d.%s" % (int(tile.y) % 1000, 
               tile.layer.extension)

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