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A308747 Number of achiral m-color cyclic compositions of n (that is, number of cyclic compositions of n with reflection symmetry where each part of size m can be colored with one of m colors).

%I A308747 #39 Feb 11 2021 21:22:27
%S A308747 1,3,6,13,23,44,73,131,210,365,575,984,1537,2611,4062,6877,10679,
%T A308747 18052,28009,47315,73386,123933,192191,324528,503233,849699,1317558,
%U A308747 2224621,3449495,5824220,9030985,15248099,23643522,39920141,61899647,104512392,162055489,273617107
%N A308747 Number of achiral m-color cyclic compositions of n (that is, number of cyclic compositions of n with reflection symmetry where each part of size m can be colored with one of m colors).
%C A308747 Cyclic compositions of a positive integer n are equivalence classes of ordered partitions of n such that two such partitions are equivalent if one can be obtained from the other by rotation. These were first studied by Sommerville (1909).
%C A308747 Symmetric cyclic compositions or circular palindromes or achiral cyclic compositions are those cyclic compositions that have at least one axis of symmetry. They were also studied by Sommerville (1909, pp. 301-304).
%C A308747 Let (c(m): m >= 1) be the input sequence and let b = (b(n): n >= 1) be the output sequence under the CPAL (circular palindrome) transform of c; that is, b(n) = (CPAC c)_n for n >= 1. Hence, b(n) is the number of symmetric cyclic compositions of n where a part of size m can be colored with one of c(m) colors. If C(x) = Sum_{m >= 1} c(m)*x^m is the g.f. of the input sequence c, then the g.f. of b = (CPAL c) is Sum_{n >= 1} b(n)*x^n = (1 + C(x))^2/(2 * (1 - C(x^2))) - (1/2).
%C A308747 For the current sequence, the input sequence is c(m) = m for m >= 1, and we are dealing with the so-called "m-color" compositions. m-color linear compositions were studied by Agarwal (2000), whereas m-color cyclic compositions were studied by Gibson (2017) and Gibson et al. (2018).
%C A308747 Thus, for the current sequence, a(n) is the number of symmetric (achiral) cyclic compositions of n where a part of size m may be colored with one of m colors (for each m >= 1).
%C A308747 The function A(x) = (exp(Pi*(x + 1)*I)*phi^(-x - 4) - exp(2*I*Pi*x)*phi^(4 - x) + exp(Pi*x*I)*phi^(x - 4) + phi^(x + 4))/sqrt(5) - 2*x, where phi is the golden ratio, shows that the sequence can be easily extended to all integers. - _Peter Luschny_, Aug 09 2020
%H A308747 A. K. Agarwal, <a href="https://web.archive.org/web/20200714215813/https://www.insa.nic.in/writereaddata/UpLoadedFiles/IJPAM/20005b15_1421.pdf">n-colour compositions</a>, Indian J. Pure Appl. Math. 31 (11) (2000), 1421-1427.
%H A308747 Christian G. Bower, <a href="/transforms2.html">Transforms (2)</a>.
%H A308747 Petros Hadjicostas, <a href="https://doi.org/10.2140/moscow.2020.9.173">Generalized colored circular palindromic compositions</a>, Moscow Journal of Combinatorics and Number Theory, 9(2) (2020), 173-186.
%H A308747 Meghann Moriah Gibson, <a href="https://digitalcommons.georgiasouthern.edu/etd/1583/">Combinatorics of compositions</a>, Master of Science, Georgia Southern University, 2017.
%H A308747 Meghann Moriah Gibson, Daniel Gray, and Hua Wang, <a href="https://doi.org/10.1016/j.disc.2018.08.001">Combinatorics of n-color compositions</a>, Discrete Mathematics 341 (2018), 3209-3226.
%H A308747 D. M. Y. Sommerville, <a href="https://doi.org/10.1112/plms/s2-7.1.263">On certain periodic properties of cyclic compositions of numbers</a>, Proc. London Math. Soc. S2-7(1) (1909), 263-313.
%F A308747 CPAL (circular palindrome) transform of 1, 2, 3, 4, ...
%F A308747 a(n) = 2*a(n - 1) + 2*a(n - 2) - 6*a(n - 3) + 2*a(n - 4) + 2*a(n - 5) - a(n - 6) for n >= 7 with a(1) = 1, a(2) = 3, a(3) = 6, a(4) = 13, a(5) = 23, and a(6) = 44.
%F A308747 a(n) = 3*a(n - 2) - a(n - 4) + 2*(n - 2) for n >= 5 with a(1) = 1, a(2) = 3, a(3) = 6, and a(4) = 13.
%F A308747 a(n) = Fib(n + 4) + (-1)^n * Fib(n - 4) - 2*n for n >= 4, where Fib(n) = A000045(n).
%F A308747 G.f.: x * (1 + x - 2*x^2 + x^3 + x^4)/((1 - x)^2 * (1 - x - x^2) * (1 + x - x^2)).
%e A308747 We have a(1) = 1 because we only have one symmetric cyclic composition of n = 1, namely 1_1 (and a part of size 1 can be colored with only one color).
%e A308747 We have a(2) = 3 because we have the following colored achiral cyclic compositions of n = 2: 2_1, 2_2, 1_1 + 1_1.
%e A308747 We have a(3) = 6 because we have the following colored achiral cyclic compositions of n = 3: 3_1, 3_2, 3_3, 1_1 + 2_1, 1_1 + 2_2, 1_1 + 1_1 + 1_1.
%e A308747 We have a(4) = 13 because we have the following colored achiral cyclic compositions of n = 4: 4_1, 4_2, 4_3, 4_4, 1_1 + 3_1, 1_1 + 3_2, 1_1 + 3_3, 2_1 + 2_1, 2_1 + 2_2, 2_2 + 2_2, 1_1 + 2_1 + 1_1, 1_1 + 2_2 + 1_1, 1_1 + 1_1 + 1_1 + 1_1.
%e A308747 We have a(5) = 23 because we have the following colored achiral cyclic compositions of n = 5:
%e A308747 (i) with one part: 5_1, 5_2, 5_3, 5_4, 5_5;
%e A308747 (ii) with two parts: 1_1 + 4_1, 1_1 + 4_2, 1_1 + 4_3, 1_1 + 4_4, 2_1 + 3_1, 2_1 + 3_2, 2_1 + 3_3, 2_2 + 3_1, 2_2 + 3_2, 2_2 + 3_3;
%e A308747 (iii) with three parts: 1_1 + 3_1 + 1_1, 1_1 + 3_2 + 1_1, 1_1 + 3_3 + 1_1, 2_1 + 1_1 + 2_1, 2_2 + 1_1 + 2_2;
%e A308747 (iv) with four parts: 1_1 + 1_1 + 2_1 + 1_1, 1_1 + 1_1 + 2_2 + 1_1 (here, the axis of symmetry passes through one of the 1's and through 2);
%e A308747 (v) with five parts: 1_1 + 1_1 + 1_1 + 1_1 + 1_1.
%Y A308747 Cf. A000045, A001906, A005594, A032198, A032287, A088305.
%K A308747 nonn
%O A308747 1,2
%A A308747 _Petros Hadjicostas_, Jun 21 2019