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A056513 Number of primitive (period n) periodic palindromic structures using a maximum of two different symbols.

%I A056513 #24 Sep 27 2019 19:45:03
%S A056513 1,1,1,1,2,3,4,7,10,14,21,31,42,63,91,123,184,255,371,511,750,1015,
%T A056513 1519,2047,3030,4092,6111,8176,12222,16383,24486,32767,49024,65503,
%U A056513 98175,131061,196308,262143,392959,524223,785910,1048575,1572256,2097151,3144702,4194162
%N A056513 Number of primitive (period n) periodic palindromic structures using a maximum of two different symbols.
%C A056513 For example, aaabbb is not a (finite) palindrome but it is a periodic palindrome. Permuting the symbols will not change the structure.
%C A056513 Number of Lyndon compositions (aperiodic necklaces of positive integers) summing to n that can be rotated to form a palindrome. - _Gus Wiseman_, Sep 16 2018
%D A056513 M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]
%H A056513 Andrew Howroyd, <a href="/A056513/b056513.txt">Table of n, a(n) for n = 0..1000</a>
%F A056513 a(n) = Sum_{d|n} mu(d)*A056503(n/d) for n > 0.
%F A056513 a(n) = Sum_{k=1..2} A285037(n, k). - _Andrew Howroyd_, Apr 08 2017
%F A056513 G.f.: 1 + (1/2)*Sum_{k>=1} mu(k)*x^k*(2 + 3*x^k)/(1 - 2*x^(2*k)) - mu(2*k)*x^(2*k)*(1 + x^(2*k))/(1 - 2*x^(4*k)). - _Andrew Howroyd_, Sep 27 2019
%e A056513 From _Gus Wiseman_, Sep 16 2018: (Start)
%e A056513 The sequence of palindromic Lyndon compositions begins:
%e A056513   (1)  (2)  (3)  (4)    (5)    (6)      (7)
%e A056513                  (112)  (113)  (114)    (115)
%e A056513                         (122)  (1122)   (133)
%e A056513                                (11112)  (223)
%e A056513                                         (11113)
%e A056513                                         (11212)
%e A056513                                         (11122)
%e A056513 (End)
%t A056513 (* b = A164090, c = A045674 *)
%t A056513 b[n_] := (1/4)*(7 - (-1)^n)*2^((1/4)*(2*n + (-1)^n - 1));
%t A056513 c[0] = 1;
%t A056513 c[n_] := c[n] = If[EvenQ[n], 2^(n/2 - 1) + c[n/2], 2^((n - 1)/2)];
%t A056513 a56503[n_] := If[OddQ[n], b[n]/2, (1/2)*(b[n] + c[n/2])];
%t A056513 a[n_] := DivisorSum[n, MoebiusMu[#] a56503[n/#]&];
%t A056513 Array[a, 45] (* _Jean-François Alcover_, Jun 29 2018, after _Andrew Howroyd_ *)
%o A056513 (PARI) a(n) = {if(n < 1, n==0, sumdiv(n, d, moebius(d)*(2 + d%2)*(2^(n/d\2)))/(4 - n%2))} \\ _Andrew Howroyd_, Sep 26 2019
%o A056513 (PARI) seq(n) = Vec(1 + (1/2)*sum(k=1, n, moebius(k)*x^k*(2 + 3*x^k)/(1 - 2*x^(2*k)) - moebius(2*k)*x^(2*k)*(1 + x^(2*k))/(1 - 2*x^(4*k)) + O(x*x^n))) \\ _Andrew Howroyd_, Sep 27 2019
%Y A056513 Row sums of A179317.
%Y A056513 Cf. A008965, A025065, A056476, A059966, A242414, A285037, A317085, A317086, A317087, A318731.
%K A056513 nonn
%O A056513 0,5
%A A056513 _Marks R. Nester_
%E A056513 a(17)-a(45) from _Andrew Howroyd_, Apr 08 2017
%E A056513 a(0)=1 prepended by _Andrew Howroyd_, Sep 27 2019