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A263104
a(3k), a(3k+1), a(3k+2) are the numbers of edges between each vertex-pair in a multigraph with 3 vertices and k edges (with at least one edge between each vertex-pair) which has the minimum number of distinct cycles, if that arrangement of edges is unique.
3
1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 2, 3, 2, 2, 3, 2, 3, 3, 1, 4, 4, 3, 3, 4, 3, 4, 4, 2, 5, 5, 4, 4, 5, 4, 5, 5, 3, 6, 6, 5, 5, 6, 5, 6, 6, 4, 7, 7, 6, 6, 7, 6, 7, 7, 5, 8, 8, 7, 7, 8, 7, 8, 8, 7, 8, 9, 8, 8, 9, 8, 9, 9, 8, 9, 10
OFFSET
9,6
COMMENTS
Cycles are counted as in A263103. Each group of three is in ascending order.
The sequence is well-defined for all n checked so far; that is, there are no known k which produce two different multigraphs with the minimal number of cycles.
Viewing the sequence in rows of three (see links) suggests that after a(64) its behavior becomes regular, with each value of a(n) appearing 9 times in 3 contiguous columns of 3 (see conjectured formula).
LINKS
Simon R. Donnelly, Python program
Eric W. Weisstein, Multigraph
FORMULA
a(n) = floor((n +4*(n%3)-1)/9) for n >= 65 (conjectured).
EXAMPLE
For k=6 there are three possible arrangements:
1,1,4: 40 cycles,
1,2,3: 28 cycles(*),
2,2,2: 33 cycles,
so a(18,19,20) = 1,2,3.
PROG
(Python) See links.
CROSSREFS
Sequence in context: A007723 A067437 A242425 * A282518 A230241 A029315
KEYWORD
nonn,walk,tabf
AUTHOR
Simon R. Donnelly, Oct 09 2015
STATUS
approved