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A106368 Necklaces with n beads of 6 colors, no 2 adjacent beads the same color. 4
6, 15, 40, 165, 624, 2635, 11160, 48915, 217040, 976887, 4438920, 20346485, 93900240, 435970995, 2034505656, 9536767665, 44878791360, 211927736135, 1003867701480, 4768372070757, 22706531350480, 108372083629275 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Vol. 226-228 (1995), 57-72.
FORMULA
CycleBG transform of (6, 0, 0, 0, ...)
CycleBG transform T(A) = invMOEBIUS(invEULER(Carlitz(A)) + A(x^2) - A) + A.
Carlitz transform T(A(x)) has g.f. 1/(1-Sum_{k>0}(-1)^(k+1)*A(x^k)).
General formula for the CycleBG transform: T(A)(x) = A(x) - Sum_{k>=0} A(x^{2k+1}) + Sum_{k>=1} (phi(k)/k)*log(Carlitz(A)(x^k)). For a proof, see the links. (For this sequence, A(x) = 6*x.) - Petros Hadjicostas, Sep 16 2017
a(n) = (1/n) * Sum_{d | n} totient(n/d) * (5*(-1)^d + 5^d) for n > 1. - Andrew Howroyd, Mar 12 2017
G.f.: m*x - Sum_{k>=1} (phi(k)/k)*((m-1)*B(x^k)+C(x^k)) = m*x - (m-1)*x/(1-x^2) - Sum_{k>=1} (phi(k)/k)*C(x^k), where B(x) = log(1+x) and C(x) = log(1-(m-1)*x). Here m=6 is the number of colors. - Petros Hadjicostas, Sep 16 2017
MATHEMATICA
a[1] = 6;
a[n_] := (1/n) Sum[EulerPhi[n/d]*(5*(-1)^d + 5^d), {d, Divisors[n]}];
Array[a, 30] (* Jean-François Alcover, Jul 02 2018, after Andrew Howroyd *)
PROG
(PARI) a(n) = if(n==1, 6, sumdiv(n, d, eulerphi(n/d)*(5*(-1)^d + 5^d))/n); \\ Andrew Howroyd, Oct 14 2017
CROSSREFS
Column 6 of A208535.
Cf. A000031.
Sequence in context: A272847 A273829 A114696 * A100491 A220030 A230165
KEYWORD
nonn
AUTHOR
Christian G. Bower, Apr 29 2005
STATUS
approved

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Last modified March 28 05:02 EDT 2024. Contains 371235 sequences. (Running on oeis4.)