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

%I A322598 #27 Dec 29 2018 07:53:29
%S A322598 1,3,8,13,20,29,39,50,64,78,94,112,131,151,174,197,222,249,277,306,
%T A322598 338,370,404,440,477,515,556,597,640,685,731,778,828,878,930,984,1039,
%U A322598 1095,1154,1213,1274,1337,1401,1466,1534,1602,1672,1744,1817
%N A322598 a(n) is the number of unlabeled rank-3 graded lattices with 3 coatoms and n atoms.
%C A322598 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.
%H A322598 Jukka Kohonen, <a href="/A322598/b322598.txt">Table of n, a(n) for n = 1..1000</a>
%H A322598 J. Kohonen, <a href="http://arxiv.org/abs/1804.03679">Counting graded lattices of rank three that have few coatoms</a>, arXiv:1804.03679 [math.CO] preprint (2018).
%H A322598 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,-1,-1,1).
%F A322598 a(n) = floor( (3/4)n^2 + (1/3)n + 1/4 ).
%F A322598 From _Colin Barker_, Dec 19 2018: (Start)
%F A322598 G.f.: x*(1 + 2*x + 4*x^2 + 2*x^3) / ((1 - x)^3*(1 + x)*(1 + x + x^2)).
%F A322598 a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6) for n>6.
%F A322598 (End)
%F A322598 From _Robert Israel_, Dec 19 2018: (Start)
%F A322598 a(6*m) = 27*m^2+2*m.
%F A322598 a(6*m+1) = 27*m^2+11*m+1.
%F A322598 a(6*m+2) = 27*m^2+20*m+3.
%F A322598 a(6*m+3) = 27*m^2+29*m+8.
%F A322598 a(6*m+4) = 27*m^2+38*m+13.
%F A322598 a(6*m+5) = 27*m^2+47*m+20.
%F A322598 These imply the conjectured G.f. and recursion.(End)
%e A322598 a(2)=3: These are the three lattices.
%e A322598     o          o          o
%e A322598    /|\        /|\        /|\
%e A322598   o o o      o o o      o o o
%e A322598   |/  |      |/_/|      |/ \|
%e A322598   o   o      o   o      o   o
%e A322598    \ /        \ /        \ /
%e A322598     o          o          o
%p A322598 seq(floor(3/4*n^2+n/3+1/4),n=1..100); # _Robert Israel_, Dec 19 2018
%t A322598 LinearRecurrence[{1, 1, 0, -1, -1, 1}, {1, 3, 8, 13, 20, 29}, 50] (* _Jean-François Alcover_, Dec 29 2018 *)
%o A322598 (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
%o A322598 (GAP) List([1..50],n->Int((3/4)*n^2+(1/3)*n+1/4)); # _Muniru A Asiru_, Dec 20 2018
%Y A322598 Third row of A300260.
%Y A322598 Next rows are A322599, A322600.
%K A322598 nonn,easy
%O A322598 1,2
%A A322598 _Jukka Kohonen_, Dec 19 2018