A117747 - OEIS

%I #17 May 21 2024 13:44:33

%S 7,15,30,74,171,444,1138,3048,8175,22427,61686,171630,479411,1347609,

%T 3801522,10768832,30595671,87190791,249085662,713268978,2046679419,

%U 5884137206,16946037930,48882597264,141215566135,408515830803,1183284759846,3431523892390

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

%C Examples of such graphs are cellular gene regulatory networks and signal transduction networks.

%C 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 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 Andrew Howroyd, <a href="/A117747/b117747.txt">Table of n, a(n) for n = 3..500</a>

%H Avi Ma'ayan, <a href="/A117747/a117747.txt">C program to produce sequence</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HelmGraph.html">Helm Graph</a>.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PlanarEmbedding.html">Planar Embedding</a>.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WebGraph.html">Web Graph</a>.

%F a(n) = 3^(n/2)/3 + (1/(2*n))*Sum_{k=0..n-1} 3^gcd(n,k) if n is even;

%F a(n) = 3^((n-1)/2)/2 + (1/(2*n))*Sum_{k=0..n-1} 3^gcd(n,k) if n is odd.

%F a(n) ~ 3^n / (2*n).

%e a(3) = 1/6 *(3^3+3^1+3^1) + 3^(2/2) / 2 = 7.

%e a(4) = 1/8 * (3^4+3^1+3^2+3^1) + 3^(4/2) / 3 = 15.

%e 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 (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 Cf. A000011.

%K nonn

%O 3,1

%A 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 Terms a(16) and beyond from _Andrew Howroyd_, Nov 07 2019