On 11/16/2012 6:30 PM, Allan Wechsler wrote:
There's a pretty Archimedian tiling with vertex figure 3.3.4.3.4 whose vertex coordination sequence has at least one of the following two properties:
1. It is not in OEIS.
2. I have miscalculated it. (This seems likely, since my "calculation" consisted of scrawling a portion of the tiling and then drawing blobs on vertices while I counted them by hand.)
I have the first five elements as 1, 5, 11, 16, 22; OEIS doesn't find anything with those elements. I'll submit this if somebody else can verify my entries.
Working by hand, I got 21 for the last term. However, it turns out to be easy to mechanize this. The network is topologically equivalent to the square grid on the integer lattice with a diagonal added at each vertex: if the coordinates are (even,even), go up and right (thus down and left for (odd,odd)); if they are (odd,even), go down and right (up and left for (even,odd)). A short Python program produced a sequence beginning 1, 5, 11, 16, 21, 27, 32, 37, 43, 48, 53, ... . The sequence of first differences has period three after the first term: 4, 6, 5, 5, 6, 5, 5, ..., (at least for the first 100 terms). I'll leave it to someone else to prove the pattern persists. -- Fred W. Helenius fredh@ix.netcom.com