OFFSET
1,2
COMMENTS
Also number of bicolored graphs, with 3 vertices in the first color class and n in the second, with no isolated vertices, and where any two vertices in one class have at most one common neighbor.
LINKS
Jukka Kohonen, Table of n, a(n) for n = 1..1000
J. Kohonen, Counting graded lattices of rank three that have few coatoms, arXiv:1804.03679 [math.CO] preprint (2018).
Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).
FORMULA
a(n) = floor( (3/4)n^2 + (1/3)n + 1/4 ).
From Colin Barker, Dec 19 2018: (Start)
G.f.: x*(1 + 2*x + 4*x^2 + 2*x^3) / ((1 - x)^3*(1 + x)*(1 + x + x^2)).
a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6) for n>6.
(End)
From Robert Israel, Dec 19 2018: (Start)
a(6*m) = 27*m^2+2*m.
a(6*m+1) = 27*m^2+11*m+1.
a(6*m+2) = 27*m^2+20*m+3.
a(6*m+3) = 27*m^2+29*m+8.
a(6*m+4) = 27*m^2+38*m+13.
a(6*m+5) = 27*m^2+47*m+20.
These imply the conjectured G.f. and recursion.(End)
EXAMPLE
a(2)=3: These are the three lattices.
o o o
/|\ /|\ /|\
o o o o o o o o o
|/ | |/_/| |/ \|
o o o o o o
\ / \ / \ /
o o o
MAPLE
seq(floor(3/4*n^2+n/3+1/4), n=1..100); # Robert Israel, Dec 19 2018
MATHEMATICA
LinearRecurrence[{1, 1, 0, -1, -1, 1}, {1, 3, 8, 13, 20, 29}, 50] (* Jean-François Alcover, Dec 29 2018 *)
PROG
(PARI) Vec(x*(1 + 2*x + 4*x^2 + 2*x^3) / ((1 - x)^3*(1 + x)*(1 + x + x^2)) + O(x^50)) \\ Colin Barker, Dec 19 2018
(GAP) List([1..50], n->Int((3/4)*n^2+(1/3)*n+1/4)); # Muniru A Asiru, Dec 20 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Jukka Kohonen, Dec 19 2018
STATUS
approved