A194091 - OEIS

%I #47 May 24 2021 23:30:55

%S 1,1,3,14,82,554,4132,33154,281459,2499523

%N The number of labeled biconnected squaregraphs with perimeter 2*n.

%C Conjecture: The next terms are 23031395, 218862682, 2134993642, 21301755122, 216752986922, 2243977451994, 23590001630055, 251411579942151. - _Michael D. Weiner_, Apr 07 2016

%C Conjecture: a(n) is also the number of connected 3-noncrossing perfect matchings on [2n] (compare to A005700). - _Michael D. Weiner_, Jun 09 2017

%H Don Knuth, <a href="http://www-cs-faculty.stanford.edu/~knuth/programs/squaregraph.w">squaregraph.w</a> (CWEB program).

%H Robert Scherer, <a href="https://www.math.ucdavis.edu/~tdenena/dissertations/202101_Scherer_Dissertation.pdf">Topics in Number Theory and Combinatorics</a>, Ph. D. Dissertation, Univ. of California Davis (2021).

%F G.f.: A(x) satisfies A(x*T(x)^2) = T(x)-1 where T(x) is the o.g.f. for A005700 (conjectured). - _Michael D. Weiner_, Jun 09 2017

%e [See A194090 for the definition of "labeled squaregraph".] For n=4 the a(4)=14 labeled biconnected squaregraphs of perimeter 8 are the straight tromino (with 4 labelings), the L tromino (with 8), and the square tetromino (with 2).

%o (CWEB) (see Knuth link).

%Y Cf. A194088, A194089, A194090, A194092, A194093.

%K nonn,hard,more

%O 1,3

%A _Don Knuth_, Aug 15 2011

%E Conjectured terms removed from data by _Michael D. Weiner_, Aug 03 2017