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A275868
Numbers n tracing out a spiral path in a pentagonal Z module thereby creating a ten-fold twin pattern with relations to quasicrystals.
1
0, 1, 2, 3, 2, 3, 4, 3, 4, 5, 4, 5, 4, 5, 6, 5, 6, 5, 6, 7, 6, 7, 6, 7, 6, 7, 8, 7, 8, 7, 8, 7, 8, 9, 8, 9, 8, 9, 8, 9, 8, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3
OFFSET
1,3
COMMENTS
Interpreted as consecutive steps along directions according to a basis of vectors represented by the tenth roots of unity in the complex number plane, the sequence traces out the path of a single spiral of a ten-fold twin pattern. All points are located on a pentagonal Z module (following the ideas of Quiquandon et al.). The ten-fold twin pattern is unique in that the local structure across the twin boundaries is identically coherent to the local structure within the twin domains. The ten-fold twin pattern is enantiomorphous, depending on the sign of the irrational shift of 1/(4*tau), with tau = (1+sqrt(5))/2 the Golden Ratio, along a [110] direction of the twin domain's orthorhombic unit cell. The sequence expresses the fact that the ten-fold twin pattern has no adjustable parameters, except for an arbitrary general scaling factor.
LINKS
W. Hornfeck, R. Kobold, M. Kolbe, D. Herlach, Quasicrystal nucleation in an intermetallic glass-former, arXiv:1410.2952 [cond-mat.mtrl-sci], 2014.
M. Quiquandon, D. Gratias, A. Sirindil, R. Portier, Merohedral twins revisited: quinary twins and beyond, Acta Cryst. A, 72 (2016), 55-61.
FORMULA
a(n) = floor(sqrt( 2*(n-1) ) + [n in { 2*k + ceiling(2*sqrt(k)) | k in N}] mod 10. Note, that floor(sqrt( 2*(n) ) is A172471 (here corrected for its offset in the combined formula), while 2*k + ceiling(2*sqrt(k)) is A078633. [] denotes the Iverson bracket.
MATHEMATICA
Table[Mod[Floor[Sqrt[2*(i-1)]]+If[MemberQ[Table[2*j+Ceiling[2*Sqrt[j]], {j, 1, i}], i], 1, 0], 10], {i, 1, 100}]
CROSSREFS
KEYWORD
easy,look,nonn
AUTHOR
Wolfgang Hornfeck, May 19 2017
STATUS
approved