cp's OEIS Frontend

A350280 Number of bracelets describing topological configurations of points and lines formed by the perpendicular bisectors of the sides of a convex cyclic n-gon.

%I A350280 #31 Jan 22 2022 21:15:09
%S A350280 0,1,1,5,9,30,69,203,519,1466,3933,11025,30345,85190,238063,671651,
%T A350280 1895265,5376856,15279117,43568435,124478129,356537150,1023113061,
%U A350280 2941713513,8472215013,24439992746,70604898953,204253079165,591631927785,1715743930880,4981202429973
%N A350280 Number of bracelets describing topological configurations of points and lines formed by the perpendicular bisectors of the sides of a convex cyclic n-gon.
%C A350280 For n>=3, a(n) is the number of topological configurations (up to cyclic shifts and reversal) of n points and n lines, where the points lie at the vertices of a convex cyclic n-gon and the lines are the perpendicular bisectors of its sides. Counting such configurations without quotienting out by cyclic shifts and reversal gives the sequence A028243.
%C A350280 a(n) is also the number of equivalence classes (up to cyclic shifts and reversal) of 2n-tuples composed of n 0's and n 1's which have an interlacing signature. The signature of a 2n-tuple (v_1,...,v_{2n}) is the n-tuple (s_1,...,s_n) defined by s_i=v_i+v_{i+n}. The signature is called interlacing if after deleting the 1's, there are letters remaining and the remaining 0's and 2's are alternating.
%H A350280 Andrew Howroyd, <a href="/A350280/b350280.txt">Table of n, a(n) for n = 1..1000</a>
%H A350280 P. Melotti, S. Ramassamy and P. Thévenin, <a href="https://arxiv.org/abs/2003.11006">Points and lines configurations for perpendicular bisectors of convex cyclic polygons</a>, arXiv:2003.11006 [math.CO], 2020.
%F A350280 From _Andrew Howroyd_, Dec 25 2021: (Start)
%F A350280 a(n) = (b(n) + (Sum_{d|n, n/d==1 (mod 2)} phi(n/d)*((3^d - (-1)^d)/2 - 2^d))/n)/2 where b(n) = 1 + 3^(n/2-1) for even n and 0 otherwise.
%F A350280 G.f.: (1/2)*(x^2/(1-x^2) + x^2/(1-3*x^2)) + (1/4)*Sum_{k>=0} phi(2*k+1)*log(B(x^(2*k+1)))/(2*k+1) where B(x) = (1+x)*(1-2*x)^2/(1-3*x).
%F A350280 (End)
%e A350280 For n=3, drawing the three perpendicular bisectors of a triangle divides the plane into 6 regions. Three of these regions contain one vertex of the triangle and the other three contain none. Up to cyclic shifts and reversal, the only possible configuration is (nonempty, nonempty, empty, empty, nonempty, empty), thus a(3)=1.
%e A350280 For n=3, the only 6-tuple (up to cyclic shift and reversal) which has interlacing signature is (1,1,0,0,1,0). Its signature is (1,2,0).
%e A350280 For n=4, the a(4)=5 equivalence classes of 8-tuples with interlacing signature are (0,1,0,1,0,1,0,1), (0,0,0,1,0,1,1,1), (0,1,0,1,0,0,1,1), (0,1,1,1,0,0,1,0) and (0,0,1,1,0,1,1,0).
%o A350280 (PARI) \\ here c(n) is up to rotations only.
%o A350280 c(n)={(n%2==0) + sumdiv(n, d, if(n/d%2==1, eulerphi(n/d)*((3^d - (-1)^d)/2 - 2^d)))/n}
%o A350280 a(n)={(c(n) + if(n%2==0, 3^(n/2-1)))/2} \\ _Andrew Howroyd_, Dec 25 2021
%o A350280 (PARI)
%o A350280 seq(n)=Vec((x^2/(1-x^2) + x^2/(1-3*x^2))/2 + sum(k=0, (n-1)\2, my(d=2*k+1); eulerphi(d)*log((1+x^d)*(1-2*x^d)^2/(1-3*x^d) + O(x*x^n))/d)/4, -n) \\ _Andrew Howroyd_, Dec 25 2021
%Y A350280 Cf. A028243.
%K A350280 nonn
%O A350280 1,4
%A A350280 _Sanjay Ramassamy_, Dec 23 2021
%E A350280 Terms a(11) and beyond from _Andrew Howroyd_, Dec 25 2021