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 A106369 Number of circular compositions of n such that no two adjacent parts are equal. 6
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS 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 LINKS 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 FORMULA 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 EXAMPLE 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. MATHEMATICA 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 *) CROSSREFS Cf. A000031, A008965, A212322, A292906. Sequence in context: A241744 A308514 A039866 * A298436 A228577 A032062 Adjacent sequences: A106366 A106367 A106368 * A106370 A106371 A106372 KEYWORD nonn AUTHOR Christian G. Bower, Apr 29 2005 EXTENSIONS Name clarified by Andrew Howroyd, Oct 12 2017 STATUS approved

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Last modified March 29 14:32 EDT 2023. Contains 361599 sequences. (Running on oeis4.)