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A322598
a(n) is the number of unlabeled rank-3 graded lattices with 3 coatoms and n atoms.
4
1, 3, 8, 13, 20, 29, 39, 50, 64, 78, 94, 112, 131, 151, 174, 197, 222, 249, 277, 306, 338, 370, 404, 440, 477, 515, 556, 597, 640, 685, 731, 778, 828, 878, 930, 984, 1039, 1095, 1154, 1213, 1274, 1337, 1401, 1466, 1534, 1602, 1672, 1744, 1817
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
J. Kohonen, Counting graded lattices of rank three that have few coatoms, arXiv:1804.03679 [math.CO] preprint (2018).
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
Third row of A300260.
Next rows are A322599, A322600.
Sequence in context: A120883 A317195 A352940 * A363034 A317194 A319128
KEYWORD
nonn,easy
AUTHOR
Jukka Kohonen, Dec 19 2018
STATUS
approved