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%I #42 Sep 27 2019 22:44:30
%S 1,1,1,2,3,5,7,12,14,27,31,54,63,119,123,240,255,490,511,990,1015,
%T 2015,2047,4020,4092,8127,8176,16254,16383,32607,32767,65280,65503,
%U 130815,131061,261576,262143,523775,524223,1047540,1048575,2096003,2097151,4192254
%N a(n) = AP(n) is the total number of aperiodic k-palindromes of n, 1 <= k <= n.
%C A k-composition of n is an ordered collection of k positive integers (parts) which sum to n.
%C A k-composition is aperiodic (primitive) if its period is k, or if it is not the concatenation of a smaller composition.
%C A k-palindrome of n is a k-composition of n which is a palindrome.
%C This sequence is AP(n), the total number of aperiodic k-palindromes of n, 1 <= k <= n.
%C For example AP(6)=5 because the number n=6
%C has 1 aperiodic 1-palindrome, namely 6 itself;
%C has 1 aperiodic 3-palindrome, namely 141;
%C has 2 aperiodic 4-palindromes, namely 2112 and 1221;
%C has 1 aperiodic 5-palindrome, namely 11211.
%C This gives a total of 1+1+2+1=5 aperiodic palindromes of 6.
%C Number of achiral set partitions of a primitive cycle of n elements having up to two different elements. - _Robert A. Russell_, Jun 19 2019
%D John P. McSorley, Counting k-compositions of n with palindromic and related structures. Preprint, 2010.
%H Andrew Howroyd, <a href="/A179781/b179781.txt">Table of n, a(n) for n = 1..200</a>
%H Hunki Baek, Sejeong Bang, Dongseok Kim, Jaeun Lee, <a href="http://arxiv.org/abs/1412.2426">A bijection between aperiodic palindromes and connected circulant graphs</a>, arXiv:1412.2426 [math.CO], 2014.
%F a(n) = Sum_{d | n} moebius(n/d)*2^(floor(d/2)) (see Baek et al. page 9). - _Michel Marcus_, Dec 09 2014
%F a(n) = 2*A000046(n) - A000048(n) = A000048(n) - 2*A308706(n) = A000046(n) - A308706(n). - _Robert A. Russell_, Jun 19 2019
%F A016116(n) = Sum_{d|n} a(d). - _Robert A. Russell_, Jun 19 2019
%F G.f.: Sum_{k>=1} mu(k)*x^k*(1 + 2*x^k)/(1 - x^(2*k)). - _Andrew Howroyd_, Sep 27 2019
%e For a(7)=7, the achiral set partitions are 0000001, 0000011, 0000101, 0000111, 0001001, 0010011, and 0010101. - _Robert A. Russell_, Jun 19 2019
%t a[n_] := DivisorSum[n, MoebiusMu[n/#] * 2^Floor[#/2]&];
%t Array[a, 44] (* _Jean-François Alcover_, Nov 04 2017 *)
%o (PARI) a(n) = sumdiv(n, d, moebius(n/d) * 2^(d\2)); \\ _Michel Marcus_, Dec 09 2014
%Y Row sums of A179519.
%Y A000048 (oriented), A000046 (unoriented), A308706 (chiral), A016116 (not primitive). - _Robert A. Russell_, Jun 19 2019
%K nonn
%O 1,4
%A _John P. McSorley_, Jul 26 2010
%E More terms from _Michel Marcus_, Dec 09 2014
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