[seqfan] Re: A200000

[Andrew Weimholt]

On Mon, Nov 14, 2011 at 9:52 AM, Alonso Del Arte
<alonso.delarte at gmail.com> wrote:
>
> What I would suggest for A200000 is something
> mathematically defined but in such a simple way that it almost seems it
> ought to be A020000 instead.
>

Well, I don't know if any of these are interesting enough for A200000, but
a few of these fall in the "I can't believe it's not already in the
OEIS" category,
and thus fit Alonso's "seems like it ought to be A020000" criteria.

a(n) = Minimum number of nodes for a graph with exactly n cycles (of any length)

    0, 3, 5, 4, 6, 8, 5, 4, 6, 8, 6, 6, 5, 5, 6, 6, 8, 7, 6, 6, 6, 6,
5, 6, 6, 7, 7, 7, 7, 7, 6, 7, 6

T(n,k), with rows containing (in ascending order) the possible values
of numbers of cycles on a graph with n nodes.

    0,
    0,
    0,
    0, 1
    0, 1, 3, 7,
    0, 1, 2, 3, 6, 7, 12, 13, 22, 37,
    0, 1, 2, 3, 4, 6, 7, 8, 10, 11, 12, 13, 14, 15, 18, 19, 20, 21,
22, 23, 24, 30, 32, 33, 35, 36, 37, 38, 39, 52, 53, 54, 59, 63, 85,
91, 133, 197

The row lengths of the above triangle are…

    1, 1, 1, 2, 4, 10, 38, 160, 830, 5223,

a(n) = smallest number, k, such that there exist no graph on n nodes
with exactly k cycles.

    1, 1, 1, 2, 2, 4, 5, 5, 268, 2389,

I also have versions of all these sequences which are restricted to
cycles of a particular length (up to k=9)

For example…

a(n) = Minimum number of nodes for a graph with exactly n 3-cycles

    0, 3, 4, 5, 4, 5, 6, 5, 6, 6, 5, 6, 6, 6, 7, 7, 6

None of these are in the OEIS yet, but I plan to submit them soon.

Andrew