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
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.
Programs
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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
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
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