A286110 Number of distinct hexaflexagons of length n.

1, 1, 1, 3, 3, 7, 8, 17, 21, 47, 63, 132, 205, 411, 685, 1353, 2385, 4643, 8496, 16430, 30735, 59343, 112531, 217245, 415628, 803209, 1545463, 2991191, 5778267, 11201883, 21702708, 42141575, 81830748, 159140895, 309590883, 602938098, 1174779397, 2290920127

Offset: 3

Author

Michel Marcus, May 02 2017

Keywords

• nonn

Links

• Marshall Hampton, Constructing and Counting Hexaflexagons, arXiv:1704.07775 [math.CO], 2017.

Crossrefs

Cf. A007282, A286111.

Programs

Maple

A286110 := proc(n)
    if type(n,'odd') then
        add(A052307(n,ceil(n/2)+1+3*i),i=0..n/6+1) ;
    else
        add(A052307(n,ceil(n/2)+3*i),i=0..n/6) ;
        %-A052307(n,n/2)/2+A007148(n/2)/2-1
    end if;
end proc:
seq(A286110(n),n=3..40) ; # R. J. Mathar, Jul 23 2017

Mathematica

A007148[n_] := (1/2)*(2^(n - 1) + Total[EulerPhi[2*#]*2^(n/#) & /@ Divisors[n]]/(2*n));
A052307[n_, k_] := Module[{hk = Mod[k, 2], a = 0}, If[k == 0, Return[1]]; Do[a = a + EulerPhi[d]*Binomial[n/d - 1, k/d - 1], {d, Divisors[GCD[k, n]]}]; (a/k + Binomial[Floor[(n - hk)/2], Floor[k/2]])/2];
a[n_] := Module[{s}, If[Mod[n, 2] == 1, Sum[A052307[n, Ceiling[n/2] + 1 + 3*i], {i, 0, Floor[n/6] + 1}], s = Sum[A052307[n, Ceiling[n/2] + 3*i], {i, 0, Floor[n/6] }]; s - A052307[n, n/2]/2 + A007148[n/2]/2 - 1]];
Table[a[n], {n, 3, 40}] (* Jean-François Alcover, Nov 28 2017, after R. J. Mathar *)

Python

from sympy import binomial as C, totient, divisors, gcd, floor, ceiling
def a007148(n):
    if n==1: return 1
    return 2**(n - 2) + sum(totient(2*d)*2**(n//d) for d in divisors(n))//(4*n)
def a052307(n, k): return 1 if n==0 else C((n//2) - k%2 * (1 - n%2), (k//2))/2 + sum(totient(d)*C(n//d, k//d) for d in divisors(gcd(n, k)))/(2*n)
def a(n):
    if n%2: return sum([a052307(n, ceiling(n/2) + 1 + 3*i) for i in range(n//6 + 2)])
    else:
        s=sum([a052307(n, ceiling(n/2) + 3*i) for i in range(n//6 + 1)])
        return s - a052307(n, n//2)//2 + a007148(n//2)//2 - 1
print([a(n) for n in range(3, 41)]) # Indranil Ghosh, Jul 24 2017, after Maple code