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

3,4

Links

Table of n, a(n) for n=3..40.

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

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 *)

Prog

(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

Crossrefs

Cf. A007282, A286111.

Sequence in context: A304179 A182559 A108046 * A116157 A056357 A288728

Adjacent sequences: A286107 A286108 A286109 * A286111 A286112 A286113

Keyword

nonn

Author

Michel Marcus, May 02 2017

Status

approved