cp's OEIS Frontend

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.

A117747 Number of different configurations of cycles (loops) in graphs containing directed and undirected links.

This page as a plain text file.
%I A117747 #17 May 21 2024 13:44:33
%S A117747 7,15,30,74,171,444,1138,3048,8175,22427,61686,171630,479411,1347609,
%T A117747 3801522,10768832,30595671,87190791,249085662,713268978,2046679419,
%U A117747 5884137206,16946037930,48882597264,141215566135,408515830803,1183284759846,3431523892390
%N A117747 Number of different configurations of cycles (loops) in graphs containing directed and undirected links.
%C A117747 Examples of such graphs are cellular gene regulatory networks and signal transduction networks.
%C A117747 a(n) is also the number of distinct planar embeddings of the n-helm and n-web graphs. - _Eric W. Weisstein_, May 21 2024
%D A117747 Ma'ayan, A., Jenkins, S. L., Neves, S., Hasseldine, A., Grace, E., Dubin-Thaler, B., Eungdamrong, N. J., Weng, G., Ram, P. T., Rice, J. J., Kershenbaum, A., Stolovitzky, G. A., Blitzer, R. D. and Iyengar, R., Formation of regulatory patterns during signal propagation in a Mammalian cellular network. Science. 2005 Aug 12;309
%H A117747 Andrew Howroyd, <a href="/A117747/b117747.txt">Table of n, a(n) for n = 3..500</a>
%H A117747 Avi Ma'ayan, <a href="/A117747/a117747.txt">C program to produce sequence</a>
%H A117747 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HelmGraph.html">Helm Graph</a>.
%H A117747 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PlanarEmbedding.html">Planar Embedding</a>.
%H A117747 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WebGraph.html">Web Graph</a>.
%F A117747 a(n) = 3^(n/2)/3 + (1/(2*n))*Sum_{k=0..n-1} 3^gcd(n,k) if n is even;
%F A117747 a(n) = 3^((n-1)/2)/2 + (1/(2*n))*Sum_{k=0..n-1} 3^gcd(n,k) if n is odd.
%F A117747 a(n) ~ 3^n / (2*n).
%e A117747 a(3) = 1/6 *(3^3+3^1+3^1) + 3^(2/2) / 2 = 7.
%e A117747 a(4) = 1/8 * (3^4+3^1+3^2+3^1) + 3^(4/2) / 3 = 15.
%e A117747 The 7 cycles of length 3 are: --> 0 --> 0 --> 0, --> 0 <-- 0 --> 0, -0 --> 0 --> 0, -0 --> 0 <-- 0, -0 <-- 0 --> 0, -0-0 --> 0, -0-0-0.
%o A117747 (PARI) a(n)={if(n%2, 3^((n-1)/2)/2, 3^(n/2-1)) + sum(k=0, k=n-1, 3^gcd(n,k))/(2*n)} \\ _Andrew Howroyd_, Nov 07 2019
%Y A117747 Cf. A000011.
%K A117747 nonn
%O A117747 3,1
%A A117747 Avi Ma'ayan (avi.maayan(AT)mssm.edu), Guillermo Cecchi, John Wagner, Ravi Rao, Azi Lipshtat, Ravi Iyengar and Gustavo Stolovitzky, Apr 28 2006
%E A117747 Terms a(16) and beyond from _Andrew Howroyd_, Nov 07 2019