

A275868


Numbers n tracing out a spiral path in a pentagonal Z module thereby creating a tenfold 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
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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 tenfold twin pattern. All points are located on a pentagonal Z module (following the ideas of Quiquandon et al.). The tenfold 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 tenfold 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 tenfold twin pattern has no adjustable parameters, except for an arbitrary general scaling factor.


LINKS

G. C. Greubel, Table of n, a(n) for n = 1..1000
W. Hornfeck, R. Kobold, M. Kolbe, D. Herlach, Quasicrystal nucleation in an intermetallic glassformer, arXiv:1410.2952 [condmat.mtrlsci], 2014.
M. Quiquandon, D. Gratias, A. Sirindil, R. Portier, Merohedral twins revisited: quinary twins and beyond, Acta Cryst. A, 72 (2016), 5561.


FORMULA

a(n) = floor(sqrt( 2*(n1) ) + [n in { 2*k + ceil(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 + ceil(2*sqrt(k)) is A078633. [] denotes the Iverson bracket.


MATHEMATICA

Table[Mod[Floor[Sqrt[2*(i1)]]+If[MemberQ[Table[2*j+Ceiling[2*Sqrt[j]], {j, 1, i}], i], 1, 0], 10], {i, 1, 100}]


CROSSREFS

Cf. A001622, A078633, A172471.
Sequence in context: A173523 A199323 A156549 * A100795 A045781 A045670
Adjacent sequences: A275865 A275866 A275867 * A275869 A275870 A275871


KEYWORD

easy,look,nonn


AUTHOR

Wolfgang Hornfeck, May 19 2017


STATUS

approved



