New Pisot tiling from Shigeki Akiyama
- From: Roger Bagula <rlbagula@xxxxxxxxxxxxx>
- Date: Fri, 11 May 2007 21:21:29 GMT
http://mathweb.sc.niigata-u.ac.jp/~akiyama/Research2.html
Shigeki Akiyama, Pisot number syetem and its dual tiling,
`Physics and Theoretical Computer Science' , ed. by J.P. Gazeau
et al., IOS Press (2007) 133-154. PDF
<http://mathweb.sc.niigata-u.ac.jp/%7Eakiyama/papers/cargese_note.pdf>
He calls his new tile a Periodic sofic tiling associated with the Pisot polynomial:
x^3-3*x^2+2*x-1
( I had named this type of Pisot as a count down because of the 3,2,1... )
NSolve[x^3 - 3*x^2 + 2*x - 1 == 0, x]
{{x -> 0.337641 - 0.56228 *I}, {x -> 0.337641 + 0.56228*I}, {x -> 2.32472}}
A095263 <http://www.research.att.com/%7Enjas/sequences/A095263> Gary Adamson:
( I don't think Gary was aware that a tile was associated with this particular Pisot)
a[1] = 1; a[2] = 3; a[3] = 7;
a[n_] := a[n] = 3a[n - 1] - 2a[n - 2] + a[n - 3];
Table[ a[n], {n, 22}]
Matrix Markov:M = {{0, 1, 0}, {0, 0, 1}, {1, -2, 3}}
v[1] = {0, 0, 1}
v[n_] := v[n] = M.v[n - 1];
a1 = Table[v[n][[1]], {n, 1, 50}]
a2 = Table[v[n][[2]], {n, 1, 50}]
a3 = Table[v[n][[3]], {n, 1, 50}]
Substition for the matrix is:
1->2
2->3
3->1,-2,-2,3,3,3
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