A263103 - OEIS

A263103

Minimum possible number of cycles without repeated edges on a multigraph with 3 vertices and n edges (when each vertex pair must have at least one edge).

%I #16 Feb 16 2025 08:33:27

%S 1,3,8,28,103,415,2176,12888,77787,585411,4616376,37165472,359048127,

%T 3547499455,35666828200,418731250432,4974631473123,59967609426243,

%U 827065614194184,11481419054280168,161364022042775615,2553658998728779999

%N Minimum possible number of cycles without repeated edges on a multigraph with 3 vertices and n edges (when each vertex pair must have at least one edge).

%C Cycles have length at least 2, and may repeat vertices but not edges.

%C Cycles p,q are equivalent if the vertex-edge sequence of q can be made by rotating or reversing that of p. The multigraphs have no loops.

%C Edge arrangements which produce these numbers are given in A263104. Maximum possible number of cycles appears to be A263102(n-1), generated by edges arranged [1,1,n-2].

%H Simon R. Donnelly, <a href="/A263103/b263103.txt">Table of n, a(n) for n = 3..92</a>

%H Simon R. Donnelly, <a href="/A263103/a263103.py.txt">Python program to compute a(n)</a>

%H Eric W. Weisstein, <a href="https://mathworld.wolfram.com/Multigraph.html">Multigraph</a>

%e For n=6 there are three possible arrangements:

%e 1,1,4: 40 cycles,

%e 1,2,3: 28 cycles,

%e 2,2,2: 33 cycles,

%e so a(6) = 28.

%o (Python) # See links.

%Y Cf. A263102, A263104.

%K nonn,walk,hard,changed

%O 3,2

%A _Simon R. Donnelly_, Oct 09 2015