%I A000086
%S A000086 1,0,1,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,2,0,2,0,0,0,0,0,0,0,0,0,2,0,0,0,0,
%T A000086 0,2,0,2,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,2,0,0,0,2,0,0,0,0,0,2,0,0,0,
%U A000086 0,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,4,0,2,0,0,0,2,0,0,0,0,0,2,0,0
%N A000086 Number of solutions to x^2 - x + 1 == 0 (mod n).
%C A000086 Number of elliptic points of order 3 for GAMMA_0 (n).
%C A000086 Equivalently, number of fixed points of GAMMA_0 (n) of type rho.
%C A000086 Values are 0 or a power of 2.
%D A000086 Fell, Harriet; Newman, Morris; Ordman, Edward; Tables of genera of groups 
               of linear fractional transformations. J. Res. Nat. Bur. Standards 
               Sect. B 67B 1963 61-68.
%D A000086 B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, 
               p. 101.
%D A000086 G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, 
               Princeton, 1971, see p. 25, Eq. (3).
%D A000086 John S. Rutherford, Sublattice enumeration. IV. Equivalence classes of 
               plane sublattices by parent Patterson symmetry and colour lattice 
               group type, Acta Cryst. (2009). A65, 156163. [See Table 4].
%H A000086 Christian G. Bower, <a href="b000086.txt">Table of n, a(n) for n=1..2000</
               a>
%F A000086 Multiplicative with a(p^e) = 1 if p = 3 and e = 1; 0 if p = 3 and e > 
               1; 2 if p == 1 (mod 3); 0 if p == 2 (mod 3). - David W. Wilson (davidwwilson(AT)comcast.net), 
               Aug 01, 2001.
%p A000086 with(numtheory); A000086 := proc (n) local d, s; if modp(n,9) = 0 then 
               RETURN(0) fi; s := 1; for d in divisors(n) do if isprime(d) then 
               s := s*(1+eval(legendre(-3,d))) fi od; s end: (Gene Smith, May 22 
               2006)
%t A000086 Array[ Function[ n, If[ EvenQ[ n ] || Mod[ n, 9 ]==0, 0, Count[ Array[ 
               Mod[ #^2-#+1, n ]&, n, 0 ], 0 ] ] ], 84 ]
%o A000086 (PARI) a(n)=if(n<1,0,sum(x=0,n-1,(x^2-x+1)%n==0))
%o A000086 (PARI) a(n)=if(n<1,0,direuler(p=2,n,if(p==3,1+X,if(p%3==2,1,(1+X)/(1-X))))[n])
%Y A000086 Cf. A000089, A000091, A001616, A014683.
%Y A000086 Sequence in context: A030201 A055668 A045839 this_sequence A045838 A045837 
               A126825
%Y A000086 Adjacent sequences: A000083 A000084 A000085 this_sequence A000087 A000088 
               A000089
%K A000086 nonn,easy,nice,mult
%O A000086 1,7
%A A000086 N. J. A. Sloane (njas(AT)research.att.com).