[math-fun] Coordination sequence for 3.3.4.3.4 lattice
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.
On 11/16/12, Allan Wechsler <acwacw@gmail.com> 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:
The Euclidean plane tiling bearing the same relation to the square lattice as the "snub cube" does to the cuboctahedron in 3-space.
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.)
A segment of the tiling is displayed at http://www.uwgb.edu/dutchs/symmetry/archtil.htm together with a nice comment about its enantiomorph ...
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.
How exactly do you define this sequence, and why is it interesting? Fred Lunnon
Also, there are more extensive download/printable segments at http://www.uwgb.edu/dutchs/symmetry/archtil.htm which might save you some drawing effort --- the 1/4 inch should be adequate for starters --- but I wouldn't recommend the 1 mm! WFL On 11/17/12, Fred lunnon <fred.lunnon@gmail.com> wrote:
On 11/16/12, Allan Wechsler <acwacw@gmail.com> 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:
The Euclidean plane tiling bearing the same relation to the square lattice as the "snub cube" does to the cuboctahedron in 3-space.
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.)
A segment of the tiling is displayed at http://www.uwgb.edu/dutchs/symmetry/archtil.htm together with a nice comment about its enantiomorph ...
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.
How exactly do you define this sequence, and why is it interesting?
Fred Lunnon
Coordination sequences are of interest to chemists and to combinatorial people. I've written a few papers abut them (see my home page). One way to define the CS of an infinite graph whose group is transitive on the vertices to let a(n) be the number of vertices whose edge distance from one fixed vertex is n. Of course there are several ways to define the graph corresponding to a "net" (cf. Wells's book) There are a lot of coordination sequences in the OEIS - look in the Index under Coordination Neil On Fri, Nov 16, 2012 at 7:58 PM, Fred lunnon <fred.lunnon@gmail.com> wrote:
On 11/16/12, Allan Wechsler <acwacw@gmail.com> 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:
The Euclidean plane tiling bearing the same relation to the square lattice as the "snub cube" does to the cuboctahedron in 3-space.
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.)
A segment of the tiling is displayed at http://www.uwgb.edu/dutchs/symmetry/archtil.htm together with a nice comment about its enantiomorph ...
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.
How exactly do you define this sequence, and why is it interesting?
Fred Lunnon
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-- Dear Friends, I have now retired from AT&T. New coordinates: Neil J. A. Sloane, President, OEIS Foundation 11 South Adelaide Avenue, Highland Park, NJ 08904, USA Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
For distance 0, ...,12 by hand I find number of vertices = 1, 5, 11, 16, 21, 27, 32, 37, 44, 48, 54, 60, 64; and not in OEIS. WFL On 11/17/12, Neil Sloane <njasloane@gmail.com> wrote:
Coordination sequences are of interest to chemists and to combinatorial people. I've written a few papers abut them (see my home page).
One way to define the CS of an infinite graph whose group is transitive on the vertices to let a(n) be the number of vertices whose edge distance from one fixed vertex is n.
Of course there are several ways to define the graph corresponding to a "net" (cf. Wells's book)
There are a lot of coordination sequences in the OEIS - look in the Index under Coordination
Neil
On Fri, Nov 16, 2012 at 7:58 PM, Fred lunnon <fred.lunnon@gmail.com> wrote:
On 11/16/12, Allan Wechsler <acwacw@gmail.com> 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:
The Euclidean plane tiling bearing the same relation to the square lattice as the "snub cube" does to the cuboctahedron in 3-space.
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.)
A segment of the tiling is displayed at http://www.uwgb.edu/dutchs/symmetry/archtil.htm together with a nice comment about its enantiomorph ...
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.
How exactly do you define this sequence, and why is it interesting?
Fred Lunnon
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Dear Friends, I have now retired from AT&T. New coordinates:
Neil J. A. Sloane, President, OEIS Foundation 11 South Adelaide Avenue, Highland Park, NJ 08904, USA Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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
FWH is right. A manual computation really needs three colours, but I could lay hands only on red and black --- couldn't blue it, so blew it ... A bull-at-the-gate proof, via separate recursions for each shape of local neighbourhood, must be straightforward but looks a little tedious. Visually however, it's easy to see what's happening: the boundary contour at distance n translates outwards in four separate continuous sections to form the contour at distance n+3, leaving 4 gaps each of length 4 at the corners, which are filled by translation from contour n-3. So the recurrence is simply f(n+3) = f(n) + 16. WFL On 11/17/12, Fred W. Helenius <fredh@ix.netcom.com> wrote:
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
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participants (4)
-
Allan Wechsler -
Fred lunnon -
Fred W. Helenius -
Neil Sloane