OFFSET
1,5
COMMENTS
Arndt's PARI code computes a(n) as the sum, divided by n, of every 3rd term in row n of L = A050186 = Möbius transform of binomials, starting with k = (1-n) mod 3 (nonnegative remainder), where k = 0 and k = n give L(n, k) = 0 and can be omitted. Cf. A053727, EXAMPLE and second PROGRAM. - M. F. Hasler, Sep 27 2018
LINKS
Robert Israel, Table of n, a(n) for n = 1..3331
J. E. Iglesias, Enumeration of polytypes of MX and MX_2 through the use of the symmetry of the Zhadanov symbol, Acta Cryst. A 62 (3) (2006) 176-194, Table 1.
J.-F. Michon, P. Ravache, On different families of invariant irreducible polynomials over F_2, Finite fields & Applications 16 (2010) 163-174.
FORMULA
a(n) = (sum_{d|n, n/d != 0 mod 3} mu(n/d)*(2^d - (-1)^d))/(3n).
EXAMPLE
Illustrating computation via L = A050186, cf. COMMENTS: a(1) = [L(1,0)] = 0. a(2) = [L(2,2)] = 0. a(3) = L(3,1)/3 = 3/3 = 1. a(4) = ([L(4,0)] + L(4,3))/4 = 4/4 = 1. a(5) = (L(5,2) + [L(5,5)])/5 = 10/5 = 2. In [...] are terms L(n,0) = L(n,n) = 0.
MAPLE
f:= proc(n) local D, d;
D:=remove(d -> (n/3/d)::integer, numtheory:-divisors(n));
add(numtheory:-mobius(n/d)*(2^d - (-1)^d), d=D)/(3*n)
end proc:
map(f, [$1..100]); # Robert Israel, Jul 14 2019
MATHEMATICA
a[n_] := Sum[If[Mod[n/d, 3] == 0, 0, MoebiusMu[n/d]*(2^d - (-1)^d)/(3n)], {d, Divisors[n]}];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Feb 02 2023 *)
PROG
(PARI)
L(n, k) = sumdiv(gcd(n, k), d, moebius(d) * binomial(n/d, k/d) );
a(n) = sum(k=0, n, if( (n+k)%3==1, L(n, k), 0 ) ) / n;
vector(55, n, a(n))
/* Joerg Arndt, Jun 28 2012 */
(PARI) A165920(n, k=(1-n)%3)=sum(i=0, (n-k)\3, A050186(n, k+3*i))\n \\ For illustration. - M. F. Hasler, Sep 30 2018
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Jean Francis Michon, Philippe Ravache (philippe.ravache(AT)univ-rouen.fr), Sep 30 2009
STATUS
approved