A322598 a(n) is the number of unlabeled rank-3 graded lattices with 3 coatoms and n atoms.

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

Author

Jukka Kohonen, Dec 19 2018

Keywords

• nonn
• easy

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.

Examples

			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
		

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).

Crossrefs

Third row of A300260.

Next rows are A322599, A322600.

Programs

GAP

List([1..50],n->Int((3/4)*n^2+(1/3)*n+1/4)); # Muniru A Asiru, Dec 20 2018

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 *)

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

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)