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Search: id:A000086
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| A000086 |
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Number of solutions to x^2 - x + 1 == 0 (mod n). |
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+0
10
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| 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, 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, 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
(list; graph; listen)
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OFFSET
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1,7
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COMMENT
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Number of elliptic points of order 3 for GAMMA_0 (n).
Equivalently, number of fixed points of GAMMA_0 (n) of type rho.
Values are 0 or a power of 2.
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REFERENCES
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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.
B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 101.
G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton, 1971, see p. 25, Eq. (3).
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].
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LINKS
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Christian G. Bower, Table of n, a(n) for n=1..2000
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FORMULA
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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.
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MAPLE
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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)
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MATHEMATICA
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Array[ Function[ n, If[ EvenQ[ n ] || Mod[ n, 9 ]==0, 0, Count[ Array[ Mod[ #^2-#+1, n ]&, n, 0 ], 0 ] ] ], 84 ]
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PROGRAM
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(PARI) a(n)=if(n<1, 0, sum(x=0, n-1, (x^2-x+1)%n==0))
(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])
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CROSSREFS
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Cf. A000089, A000091, A001616, A014683.
Adjacent sequences: A000083 A000084 A000085 this_sequence A000087 A000088 A000089
Sequence in context: A030201 A055668 A045839 this_sequence A045838 A045837 A126825
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KEYWORD
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nonn,easy,nice,mult
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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