This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A365857 #30 Mar 09 2024 11:35:36
%S A365857 1,2,4,7,14,30,63,142,328,765,1810,4340,10461,25414,62074,152287,
%T A365857 375166,927554,2300347,5721044,14264308,35646311,89264834,223959710,
%U A365857 562878429,1416953362,3572233420,9018211989,22795835726,57690911720,146164582455,370705552702,941109975022,2391391374017,6081865318124
%N A365857 Number of cyclic compositions of 2*n into odd parts.
%C A365857 Even bisection of A032189.
%C A365857 Also the number of cyclic compositions into an even number of odd parts; because such a sum must be even, alternating terms are zero and have been removed.
%C A365857 Also the number of dual classes of cyclic n-color compositions of n. A cyclic composition is a sum of positive integers in which the order of the parts is considered up to cyclic permutation. In other words, it is the collection of components remaining in the cycle graph C_n on n vertices when one or more edges are removed, and rotations are considered equivalent. In an n-color composition, each part of size k is assigned one of k "colors" which may be represented graphically by marking one vertex in the part. (See A032198 for the number of cyclic n-color compositions.) The dual of a cyclic n-color composition is obtained by switching the roles of edges and vertices in C_n, then removing each edge that came from a previously marked vertex while marking each vertex that came from a previously removed edge. Each cyclic n-color composition of n either belongs to a dual pair or is self-dual. (See A365859 for the number of self-dual cyclic n-color compositions.)
%H A365857 A. K. Agarwal, <a href="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 A365857 Joshua P. Bowman, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL27/Bowman/bowman4.html">Compositions with an Odd Number of Parts, and Other Congruences</a>, J. Int. Seq (2024) Vol. 27, Art. 24.3.6. See p. 25.
%H A365857 Meghann Moriah Gibson, Daniel Gray, and Hua Wang, <a href="https://doi.org/10.1016/j.disc.2018.08.001">Combinatorics of n-color cyclic compositions</a>, Discrete Mathematics 341 (2018), 3209-3226.
%F A365857 G.f.: (1/2)*(Sum_{k>=1} phi(k)/k * log((1-2*x^k+x^(2*k))/(1-3*x^k+x^(2*k))) + Sum_{m>=1} phi(2*m)/(2*m) * log((1+x^m-x^(2*m))/(1-x^m-x^(2*m)))).
%F A365857 a(n) = (1/(2*n)) * Sum_{k divides 2*n} phi(k)*A001350((2*n)/k).
%F A365857 a(n) = (A032198(n) + A365859(n))/2.
%o A365857 (PARI)
%o A365857 N=99; x='x+O('x^N); B(x)=x/(1-x^2);
%o A365857 A=Vec(sum(k=1, N, eulerphi(k)/k*log(1/(1-B(x^k)))));
%o A365857 vector(#A\2,n,A[2*n]) \\ _Joerg Arndt_, Sep 22 2023
%o A365857 (Python)
%o A365857 from sympy import totient, lucas, divisors
%o A365857 def A365857(n): return sum(totient((n<<1)//k)*(lucas(k)-((k&1^1)<<1)) for k in divisors(n<<1,generator=True))//n>>1 # _Chai Wah Wu_, Sep 23 2023
%Y A365857 Cf. A001350, A032189, A032198, A365858, A365859.
%K A365857 nonn
%O A365857 1,2
%A A365857 _Joshua P. Bowman_, Sep 20 2023