OFFSET
0,3
COMMENTS
Almost cubic graphs are cubic graphs (A002829) where 2 points have degree 2 and these 2 points are non-adjacent. All other points have degree 3. They are constructed by removing an edge from the cubic graphs.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..100
N. C. Wormald, Enumeration of labelled graphs II: cubic graphs with a given connectivity, J. Lond Math Soc s2-20 (1979) 1-7, e.g.f. a(x).
FORMULA
a(n) = 3*n*A002829(n). [Wormald eq. (2.1)]
EXAMPLE
There is 1 unlabeled almost cubic graph on 4 nodes (the kite, obtained by removing an edge of the tetrahedron K_4). This has 6 = binomial(4,2) labeled versions obtained by selecting two out of 4 labels for the points of degree 2.
MATHEMATICA
terms = 14; egf = HypergeometricPFQ[{1/6, 5/6}, {}, 12x/(x^2 + 8x + 4)^(3/2)] Exp[-Log[1/4 x^2 + 2x + 1]/4 - x/3 + (x^2 + 8x + 4)^(3/2)/(24 x) - 1/(3x) - x^2/24 - 1] + O[x]^terms;
CoefficientList[egf, x](2 Range[0, terms-1])! 3 Range[0, terms-1] (* Jean-François Alcover, Nov 23 2018, from A002829 *)
PROG
(PARI) b(n) = sum(i=0, 2*n, sum(k=0, min(floor((3*n-i)/3), floor((2*n-i)/2)), sum(j=0, min(floor((3*n-i-3*k)/2), floor((2*n-i-2*k)/2)), ((-1)^(i+j)*(2*n)!*(2*(3*n-i-2*j-3*k))!)/(2^(5*n-i-2*j-4*k)*3^(2*n-i-2*j-k)*(3*n-i-2*j-3*k)!*i!*j!*k!*(2*n-i-2*j-2*k)!)))); \\ A002829
vector(20, n, n--; 3*n*b(n)) \\ Michel Marcus, Nov 10 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
R. J. Mathar, Nov 09 2018
EXTENSIONS
Terms a(11) and beyond from Andrew Howroyd, Nov 09 2018
STATUS
approved