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 A286110 Number of distinct hexaflexagons of length n. 2
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 3,4 LINKS 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): 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(floor(n/2) - k%2 * (1 - n%2), floor(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==1: return sum([a052307(n, ceiling(n/2) + 1 + 3*i) for i in xrange(floor(n/6) + 2)])     else:         s=sum([a052307(n, ceiling(n/2) + 3*i) for i in xrange(floor(n/6) + 1)])         return s - a052307(n, n/2)/2 + a007148(n/2)/2 - 1 print map(a, xrange(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

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Last modified May 20 03:10 EDT 2019. Contains 323412 sequences. (Running on oeis4.)