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 A027376 Number of ternary irreducible monic polynomials of degree n; dimensions of free Lie algebras. 34
 1, 3, 3, 8, 18, 48, 116, 312, 810, 2184, 5880, 16104, 44220, 122640, 341484, 956576, 2690010, 7596480, 21522228, 61171656, 174336264, 498111952, 1426403748, 4093181688, 11767874940, 33891544368, 97764009000, 282429535752, 817028131140, 2366564736720, 6863037256208, 19924948267224, 57906879556410 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Number of Lyndon words of length n on {1,2,3}. A Lyndon word is primitive (not a power of another word) and is earlier in lexicographic order than any of its cyclic shifts. - John W. Layman, Jan 24 2006 Exponents in an expansion of the Hardy-Littlewood constant product(1-(3*p-1)/(p-1)^3, p prime >= 5), whose decimal expansion is in A065418: the constant equals product_{n>=2} (zeta(n)*(1-2^-n)*(1-3^-n))^-a(n). - Michael Somos, Apr 05 2003 Number of aperiodic necklaces with n beads of 3 colors. - Herbert Kociemba, Nov 25 2016 REFERENCES E. R. Berlekamp, Algebraic Coding Theory, McGraw-Hill, NY, 1968, p. 84. M. Lothaire, Combinatorics on Words. Addison-Wesley, Reading, MA, 1983, p. 79. LINKS Seiichi Manyama, Table of n, a(n) for n = 0..2102 (terms 0..200 from T. D. Noe) E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665. Veronika Irvine, Lace Tessellations: A mathematical model for bobbin lace and an exhaustive combinatorial search for patterns, PhD Dissertation, University of Victoria, 2016. See Table A.2. Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1. G. Niklasch, Some number theoretical constants: 1000-digit values [Cached copy] G. Viennot, Algèbres de Lie Libres et Monoïdes Libres, Lecture Notes in Mathematics 691, Springer Verlag 1978. FORMULA a(n) = (1/n)*Sum_{d|n} mu(d)*3^(n/d). (1-3*x) = Product_{n>0} (1-x^n)^a(n). G.f.: k=3, 1 - Sum_{i>=1} mu(i)*log(1 - k*x^i)/i. - Herbert Kociemba, Nov 25 2016 a(n) ~ 3^n / n. - Vaclav Kotesovec, Jul 01 2018 EXAMPLE For n = 2 the a(2)=3 polynomials are  x^2+1, x^2+x+2, x^2+2*x+2. - Robert Israel, Dec 16 2015 MAPLE A027376 := proc(n) local d, s; if n = 0 then RETURN(1); else s := 0; for d in divisors(n) do s := s+mobius(d)*3^(n/d); od; RETURN(s/n); fi; end; MATHEMATICA a[0]=1; a[n_] := Module[{ds=Divisors[n], i}, Sum[MoebiusMu[ds[[i]]]3^(n/ds[[i]]), {i, 1, Length[ds]}]/n] a[0]=1; a[n_] := DivisorSum[n, MoebiusMu[n/#]*3^#&]/n; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Dec 01 2015 *) mx=40; f[x_, k_]:=1-Sum[MoebiusMu[i] Log[1-k*x^i]/i, {i, 1, mx}]; CoefficientList[Series[f[x, 3], {x, 0, mx}], x] (* Herbert Kociemba, Nov 25 2016 *) PROG (PARI) a(n)=if(n<1, n==0, sumdiv(n, d, moebius(n/d)*3^d)/n) CROSSREFS Column 3 of A074650. Cf. A000031, A001037, A001693, A001867, A027375, A027377, A054718, A102660. Sequence in context: A059197 A049974 A049972 * A190659 A202536 A038068 Adjacent sequences:  A027373 A027374 A027375 * A027377 A027378 A027379 KEYWORD nonn,nice,easy AUTHOR STATUS approved

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Last modified December 11 17:38 EST 2018. Contains 318049 sequences. (Running on oeis4.)