A031358 Number of coincidence site lattices of index 4n+1 in lattice Z^2.

1, 2, 0, 2, 2, 0, 2, 2, 0, 2, 2, 0, 0, 2, 0, 2, 4, 0, 2, 0, 0, 4, 2, 0, 2, 2, 0, 2, 2, 0, 0, 2, 0, 0, 2, 0, 4, 2, 0, 2, 0, 0, 2, 2, 0, 2, 4, 0, 2, 2, 0, 4, 0, 0, 0, 4, 0, 2, 2, 0, 2, 0, 0, 0, 2, 0, 4, 2, 0, 2, 2, 0, 2, 2, 0, 0, 4, 0, 2, 2, 0, 4, 0, 0, 2, 0, 0, 2, 2, 0, 0, 4, 0, 2, 4, 0, 0, 2, 0, 2, 2, 0, 2, 0, 0, 2

Offset

0,2

Links

Andrey Zabolotskiy, Table of n, a(n) for n = 0..999

M. Baake, Solution of the coincidence problem in dimensions d <= 4, in R. V. Moody, ed., Math. of Long-Range Aperiodic Order, Kluwer 1997, pp. 9-44; arXiv:math/0605222 [math.MG], 2006.

Michael Baake and Peter A. B. Pleasants, Algebraic solution of the coincidence problem in two and three dimensions, Zeitschrift für Naturforschung A 50.8 (1995): 711-717. See annotated scan of page 713.

Formula

Dirichlet series: Product_{primes p == 1 mod 4} (1+p^(-s))/(1-p^(-s)).

a(n) = 2*A106594(n) for n > 0. - Andrey Zabolotskiy, Jan 30 2020

Prog

(PARI) t1=direuler(p=2, 1200, (1+(p%4<2)*X))
t2=direuler(p=2, 1200, 1/(1-(p%4<2)*X))
t3=dirmul(t1, t2)
t4=vector(200, n, t3[4*n+1]) (and then prepend 1)

Crossrefs

Cf. A175647, A031359, A331140, A106594, A094178 (positions of nonzero terms).

Sequence in context: A347730 A329491 A123063 * A279103 A318734 A029317

Adjacent sequences: A031355 A031356 A031357 * A031359 A031360 A031361

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

More terms from N. J. A. Sloane, Mar 13 2009

Added condition that p must be prime to the Dirichlet series. - N. J. A. Sloane, May 26 2014

Offset corrected by Andrey Zabolotskiy, Jan 30 2020

Status

approved