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A106369
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Number of circular compositions of n such that no two adjacent parts are equal.
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6
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1, 1, 2, 2, 3, 6, 7, 11, 18, 29, 42, 73, 111, 183, 299, 491, 796, 1333, 2188, 3652, 6073, 10155, 16959, 28500, 47813, 80508, 135621, 228967, 386749, 654535, 1108353, 1879478, 3189495, 5418556, 9212099, 15676275, 26694509, 45493327, 77580915
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OFFSET
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1,3
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COMMENTS
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By "circular compositions" here we mean equivalence classes of compositions with parts on a circle such that two compositions are equivalent if one is a cyclic shift of the other. - Petros Hadjicostas, Oct 15 2017
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LINKS
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Vaclav Kotesovec, Table of n, a(n) for n = 1..1000
P. Hadjicostas, Cyclic, dihedral and symmetric Carlitz compositions of a positive integer, Journal of Integer Sequences, 20 (2017), Article 17.8.5.
Petros Hadjicostas, Proof of an explicit formula for Bower's CycleBG transform
Index entries for sequences related to necklaces
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FORMULA
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CycleBG transform of (1, 1, 1, 1, ...).
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)).
G.f.: x/(1-x) - Sum_{s>=1} (phi(s)/s)*f(x^s), where f(x) = log(1 - Sum_{n>=1} x^n/(1 + x^n)) + Sum_{n>=1} log(1 + x^n) and phi(s)=A000010 is Euler's totient function. - Petros Hadjicostas, Sep 06 2017
Conjecture: a(n) ~ A241902^n / n. - Vaclav Kotesovec, Sep 06 2017
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 above. (For this sequence, A(x) = x/(1-x).) - Petros Hadjicostas, Oct 08 2017
G.f.: -Sum_{s>=1} x^(2s+1)/(1-x^(2s+1)) - Sum_{s>=1} (phi(s)/s)*g(x^s), where g(x) = log(1 + Sum_{n>=1} (-x)^n/(1 - x^n)). (This formula can be proved from the general formula for the CycleBG transform given above.) - Petros Hadjicostas, Oct 10 2017
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EXAMPLE
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a(6) = 6 because the 6 circular compositions of 6: 6, 5+1, 4+2, 3+2+1, 3+1+2, 2+1+2+1.
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MATHEMATICA
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nmax = 40; Rest[CoefficientList[Series[x/(1-x) - Sum[EulerPhi[s]/s*(Log[1 - Sum[x^(s*n)/(1 + x^(s*n)), {n, 1, nmax}]] + Sum[Log[1 + x^(s*n)], {n, 1, nmax}]), {s, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Sep 06 2017, after Petros Hadjicostas *)
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CROSSREFS
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Cf. A000031, A008965, A212322, A292906.
Sequence in context: A241744 A308514 A039866 * A298436 A228577 A032062
Adjacent sequences: A106366 A106367 A106368 * A106370 A106371 A106372
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KEYWORD
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nonn
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AUTHOR
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Christian G. Bower, Apr 29 2005
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EXTENSIONS
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Name clarified by Andrew Howroyd, Oct 12 2017
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STATUS
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approved
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