A031364 Number of coincidence site modules of index 10n+1 in an icosahedral module.

1, 0, 0, 5, 6, 0, 0, 0, 10, 0, 24, 0, 0, 0, 0, 20, 0, 0, 40, 30, 0, 0, 0, 0, 30, 0, 0, 0, 60, 0, 64, 0, 0, 0, 0, 50, 0, 0, 0, 0, 84, 0, 0, 120, 60, 0, 0, 0, 50, 0, 0, 0, 0, 0, 144, 0, 0, 0, 120, 0, 124, 0, 0, 80, 0, 0, 0, 0, 0, 0, 144, 0, 0, 0, 0, 200, 0, 0

Offset

1,4

Comments

a(n) is nonzero iff n is of the form x^2+x*y+y^2 (A031363).

References

Michael Baake, "Solution of coincidence problem in dimensions d<=4", in R. V. Moody, ed., Math. of Long-Range Aperiodic Order, Kluwer 1997, pp. 9-44.

Links

Table of n, a(n) for n=1..78.

M. Baake, Solution of the coincidence problem in dimensions d <= 4, arxiv:math/0605222 (2006), Prop. 5.4.

Michael Baake and Peter AB Pleasants, Algebraic solution of the coincidence problem in two and three dimensions, Zeitschrift für Naturforschung A 50.8 (1995): 711-717. See page 716.

Sean A. Irvine, Java program (github)

Formula

Dirichlet series: ((1+5^(-s))/(1-5^(1-s))) * Product_{p = +-2 (mod 5)} ((1+p^(-2*s))/(1-p^(2*(1-s)))) * Product_{p = +-1 (mod 5)} ((1+p^(-s))/(1-p^(1-s)))^2. - Sean A. Irvine, Apr 29 2020

Crossrefs

Cf. A031363.

Sequence in context: A102060 A102058 A231409 * A078473 A215833 A110800

Adjacent sequences: A031361 A031362 A031363 * A031365 A031366 A031367

Keyword

nonn,easy

Author

N. J. A. Sloane.

Extensions

Missing a(8)=0 and more terms from Sean A. Irvine, Apr 29 2020

Status

approved