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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 (list; graph; refs; listen; history; text; internal format)
 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 Third row of A300260. Next rows are A322599, A322600. Sequence in context: A303592 A120883 A317195 * A317194 A319128 A094110 Adjacent sequences:  A322595 A322596 A322597 * A322599 A322600 A322601 KEYWORD nonn,easy AUTHOR Jukka Kohonen, Dec 19 2018 STATUS approved

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Last modified January 21 10:25 EST 2022. Contains 350477 sequences. (Running on oeis4.)