|
%I #25 Jan 31 2020 07:15:08
%S 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,
%T 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,
%U 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
%N Number of coincidence site lattices of index 4n+1 in lattice Z^2.
%H Andrey Zabolotskiy, <a href="/A031358/b031358.txt">Table of n, a(n) for n = 0..999</a>
%H 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:<a href="https://arxiv.org/abs/math/0605222">math/0605222</a> [math.MG], 2006.
%H Michael Baake and Peter A. B. Pleasants, <a href="https://doi.org/10.1515/zna-1995-0802">Algebraic solution of the coincidence problem in two and three dimensions</a>, Zeitschrift für Naturforschung A 50.8 (1995): 711-717. See annotated scan of <a href="/A031358/a031358.pdf">page 713</a>.
%F Dirichlet series: Product_{primes p == 1 mod 4} (1+p^(-s))/(1-p^(-s)).
%F a(n) = 2*A106594(n) for n > 0. - _Andrey Zabolotskiy_, Jan 30 2020
%o (PARI) t1=direuler(p=2,1200,(1+(p%4<2)*X))
%o t2=direuler(p=2,1200,1/(1-(p%4<2)*X))
%o t3=dirmul(t1,t2)
%o t4=vector(200,n,t3[4*n+1]) (and then prepend 1)
%Y Cf. A175647, A031359, A331140, A106594, A094178 (positions of nonzero terms).
%K nonn,easy,nice
%O 0,2
%A _N. J. A. Sloane_
%E More terms from _N. J. A. Sloane_, Mar 13 2009
%E Added condition that p must be prime to the Dirichlet series. - _N. J. A. Sloane_, May 26 2014
%E Offset corrected by _Andrey Zabolotskiy_, Jan 30 2020
|
