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 A377573 #24 Nov 06 2024 23:44:45
%S A377573 1,0,1,0,3,0,10,1,35,9,126,55,462,286,1717,1365,6451,6188,24463,27132,
%T A377573 93518,116281,360031,490337,1394582,2043275,5430530,8439210,21242341,
%U A377573 34621041,83411715,141290436,328589491,574274008,1297937234,2326683921,5138431851
%N A377573 Cogrowth sequence for the 14-element dihedral group D7 = <S,T | S^7, T^2, (ST)^2>.
%C A377573 Taking the overlay of the two generating functions in the bisections A072844 and A072266, shows that a(n) = A094052(n-1), n>0. - _R. J. Mathar_, Nov 05 2024
%H A377573 Sean A. Irvine, <a href="/A377573/b377573.txt">Table of n, a(n) for n = 0..1000</a>
%H A377573 H. S. M. Coxeter and W. O. J. Moser, <a href="http://doi.org/10.1007/978-3-662-21943-0">Generators and Relations for Discrete Groups</a>, 4th ed., Springer-Verlag, NY, reprinted 1984, p. 134.
%H A377573 Haggai Liu, <a href="http://www.koutschan.de/conf/ACA21/ACA2021_slides_Liu.pdf">Enumerative Properties of Cogrowth Series on Free Products of Finite Groups</a>, ACA 2021 Session on Algorithmic Combinatorics, 2021.
%H A377573 Sean A. Irvine, <a href="https://github.com/archmageirvine/joeis/blob/master/src/irvine/oeis/a377/A377573.java">Java program</a> (github)
%F A377573 G.f.: F_7(x) where F_n(x) = 1/2 + (1/(2*n)) * Sum_{j=0..n-1} 1 / (1 - 2*cos(2*Pi*j/n)*x).
%e A377573 a(4) = 3 corresponds to the TTTT = TSTS = STST = 1. Note: TSTS = (TSTS)(TT) = T(STST)T = TT = 1.
%e A377573 a(9) = 9 corresponds to the words SSSSSSSTT = SSSSSSTTS = SSSSSTTSS = SSSSTTSSS = SSSTTSSSS = SSTTSSSSS = STTSSSSSS = TTSSSSSSS = TSSSSSSST = 1.
%Y A377573 Bisections: A072266, A072844.
%Y A377573 Cf. A052964 (D5), A007583 (D6), A007582 (D8).
%K A377573 nonn,easy
%O A377573 0,5
%A A377573 _Sean A. Irvine_, Nov 01 2024