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 A056493 Number of primitive (period n) periodic palindromes using a maximum of two different symbols. 7
 2, 1, 2, 3, 6, 7, 14, 18, 28, 39, 62, 81, 126, 175, 246, 360, 510, 728, 1022, 1485, 2030, 3007, 4094, 6030, 8184, 12159, 16352, 24381, 32766, 48849, 65534, 97920, 131006, 196095, 262122, 392364, 524286, 785407, 1048446, 1571310, 2097150, 3143497 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS For example, aaabbb is not a (finite) palindrome but it is a periodic palindrome. Also number of aperiodic necklaces (Lyndon words) with two colors that are the same when turned over. REFERENCES M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for a pdf file of Chap. 2] LINKS Table of n, a(n) for n=1..42. Index entries for sequences related to Lyndon words FORMULA Sum_{d|n} mu(d)*b(n/d), where b(n) = A029744(n+1). [Corrected by Petros Hadjicostas, Oct 15 2017. The original formula referred to a previous version of sequence A029744 that had a different offset.] More generally, let gf(k) be the g.f. for the number of necklaces with reflectional symmetry but no rotational symmetry and beads of k colors. Then gf(k): Sum_{n >= 1} mu(n)*Sum_{i=0..2} binomial(k,i)*x^(n*i)/(1 - k*x^(2*n)). - Herbert Kociemba, Nov 29 2016 G.f.: Sum_{n >= 1} mu(n)*x^n*(2 + 3*x^n)/(1 - 2*x^(2*n)). The g.f. by _Herbet Kociemba_ above, with k = 2, becomes Sum_{n>=1} mu(n)*(x^n + 1)^2/(1 - 2*x^(2*n)). The two formulae differ by the "undetermined" constant Sum_{n >= 1} mu(n). - Petros Hadjicostas, Oct 15 2017 EXAMPLE a(1) = 2 with aaa... and bbb..., a(2) = 1 with ababab..., a(3) = 2 with aabaab... and abbabb..., a(4) = 3 with aaabaaab... and aabbaabb... and abbbabbb.... - Michael Somos, Nov 29 2016 MATHEMATICA mx=40; gf[x_, k_]:=Sum[ MoebiusMu[n]*Sum[Binomial[k, i]x^(n i), {i, 0, 2}]/( 1-k x^(2n)), {n, mx}]; CoefficientList[Series[gf[x, 2], {x, 0, mx}], x] (* Herbert Kociemba, Nov 29 2016 *) CROSSREFS Column 2 of A284856. Cf. A029744, A056391, A056458. Sequence in context: A097719 A249050 A341649 * A289352 A277619 A001371 Adjacent sequences: A056490 A056491 A056492 * A056494 A056495 A056496 KEYWORD nonn AUTHOR Marks R. Nester EXTENSIONS More terms and additional comments from Christian G. Bower, Jun 22 2000 STATUS approved

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Last modified July 20 16:01 EDT 2024. Contains 374459 sequences. (Running on oeis4.)