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
Empirical: a(n) == A001615(n) (mod 3) for all natural numbers n. - John M. Campbell, Apr 01 2018
From Jianing Song, Jul 03 2018: (Start)
The comment above is true. Since both a(n) and A001615(n) are multiplicative we just have to verify that for prime powers. Note that A001615(p^e) = (p+1)*p^(e-1). For p == 1 (mod 3), p+1 == 2 (mod 3) so (p+1)*p^(e-1) == 2 (mod 3); for p == 2 (mod 3), p+1 is a multiple of 3 so (p+1)*p^(e-1) == 0 (mod 3). For p = 3, if e = 1 then p+1 == 1 (mod 3); if e > 1 then (p+1)*p^(e-1) == 0 (mod 3).
Equivalently, number of solutions to x^2 + x + 1 == 0 (mod n). (End)
REFERENCES
Bruno Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 101.
Goro Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton, 1971, see p. 25, Eq. (3).
LINKS
Christian G. Bower, Table of n, a(n) for n = 1..2000
Harriet Fell, Morris Newman, and Edward Ordman, Tables of genera of groups of linear fractional transformations, J. Res. Nat. Bur. Standards Sect. B 67B (1963), 61-68.
Lorenz Halbeisen and Norbert Hungerbuehler, Number theoretic aspects of a combinatorial function, Notes on Number Theory and Discrete Mathematics 5(4) (1999), 138-150; see Definition 7 for the shadow transform.
John S. Rutherford, Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type, Acta Cryst. A65 (2009), 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
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*sqrt(3)/(3*Pi) = 0.367552... (A165952). - Amiram Eldar, Oct 11 2022
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 *)
a[n_] := a[n] = Product[{p, e} = pe; Which[p==1 || p==3 && e==1, 1, p==3 && e>1, 0, Mod[p, 3]==1, 2, Mod[p, 3]==2, 0, True, a[p^e]], {pe, FactorInteger[n]}]; Array[a, 105] (* Jean-François Alcover, Oct 18 2018 *)
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
KEYWORD
nonn,easy,nice,mult
AUTHOR
STATUS
approved