Sorry, format problem for New Sequence group generator

[N. J. A. Sloane]

    Dear Max.

    I wrote : 

    a(n) = Product_{v_i} m_i
         + Sum_{c_j} (se_j - 1)*Product_{v_k E (G_n-c_j)} m_k
          where : 
                v_i E V_n, G_n={V_n,E_n}, "E" means element
                m_i means number of matching of incident edges of v_i
                c_j means cycles in G_n
                se_j means number of start-end points in c_j
                v_k E G_n and not(v_k E c_j)
                m_k means number of matching of incident edges of v_k

    But it is not correct.
    The correct formula is the following.

    a(n) = Product_{v_i} m_i
         + Sum_{c_j} (se_j - 1)*(Product_{v_k E (G_n-c_j)} m_k - {number of partitions of (G_n-c_i) which has cycles})

    The first formula says : 

    a(4) = 3^6 + 2*(3^4-2) - 2*(3^2-1) + 2*(3^2-1)  - 1 -2*1 + 3*3^2 + 2*1 + 2*3 + 3
    = 922

    But the correct formula says : 

    second term is case of one 4-cycle
    ._. ._._._.
    |_|+._|_|_|

    = {number of synmetry}*(se_j - 1)*(Product_{v_k E G_n-c_i)} m_k - {number of partitions of (G_n-c_i) which have  {14c 3rd cell} and {14c 4th cell} and {16c 3rd+4th cell}}
    = 2*(2-1)*(3^4-1-9-1)

    I forgot the last one 6-cycle.

    And the other terms are the same as the first formula.
    Hence,

    a(4) = 922 - 2*1*9
         = 904

    It is the same as yours.

    I think that my formula is not so good.
    Because if n is not small then a computing the {number of partitions of (G_n-c_i) which has cycles}  becomes difficult.



    I will soon submit an edited A131709 to OEIS.



    Yasutoshi
    



There appears to be an error in A100729: I believe A100729(3) should be 26
rather than 25, and at least my program finds the other listed values for
each of A100729(n) and A100730(n) for n in 2..7 (and working on n=8).

It might also be useful a) to mention Ulam sequences in the comments for
these, as a phrase that might be easier to search for, and b) to link to
Finch(1992) as <http://www.expmath.org/expmath/volumes/1/1.html> rather
than just the top of the website, in both sequences.

If there is no objection to my correction I'll submit a suggested change
to Neil.

Hugo