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A000086 Number of solutions to x^2 - x + 1 == 0 (mod n). 23
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; refs; listen; history; text; internal format)
OFFSET

1,7

COMMENTS

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.

Shadow transform of central polygonal numbers A002061. - Michel Marcus, Jun 06 2013

a(A226946(n)) = 0; a(A034017(n)) > 0. - Reinhard Zumkeller, Jun 23 2013

REFERENCES

G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton, 1971, see p. 25, Eq. (3).

B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 101.

LINKS

Christian G. Bower, Table of n, a(n) for n=1..2000

Harriet Fell, Morris Newman, Edward Ordman, Tables of genera of groups of linear fractional transformations, J. Res. Nat. Bur. Standards Sect. B 67B 1963 61-68.

John S. Rutherford, Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type, Acta Cryst. (2009). A65, 156-163. [See Table 4].

N. J. A. Sloane, Transforms

FORMULA

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, Aug 01 2001

a(2*n) = a(3*n + 2) = a(9*n) = a(9*n + 6) = 0. - Michael Somos, Aug 14 2015

EXAMPLE

G.f. = x + x^3 + 2*x^7 + 2*x^13 + 2*x^19 + 2*x^21 + 2*x^31 + 2*x^37 + 2*x^39 + ...

MAPLE

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 Ward Smith, May 22 2006

MATHEMATICA

Array[ Function[ n, If[ EvenQ[ n ] || Mod[ n, 9 ]==0, 0, Count[ Array[ Mod[ #^2-#+1, n ]&, n, 0 ], 0 ] ] ], 84 ]

a[ n_] := If[ n < 1, 0, Length[ Select[ (#^2 - # + 1)/n & /@ Range[n], IntegerQ]]]; (* Michael Somos, Aug 14 2015 *)

PROG

(PARI) {a(n) = if( n<1, 0, sum( x=0, n-1, (x^2 - x + 1)%n==0))}; /* Nov 15 2002 */

(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])}; /* Nov 15 2002 */

(Haskell)

a000086 n = if n `mod` 9 == 0 then 0

  else product $ map ((* 2) . a079978 . (+ 2)) $ a027748_row $ a038502 n

-- Reinhard Zumkeller, Jun 23 2013

CROSSREFS

Cf. A000089, A000091, A001616, A014683.

Cf. A027748, A079978, A038502, A007949.

Sequence in context: A030201 A055668 A045839 * A045838 A293814 A045837

Adjacent sequences:  A000083 A000084 A000085 * A000087 A000088 A000089

KEYWORD

nonn,easy,nice,mult

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified November 21 16:27 EST 2017. Contains 295003 sequences.