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A060488
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Number of 4-block ordered tricoverings of an unlabeled n-set.
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7
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4, 13, 28, 50, 80, 119, 168, 228, 300, 385, 484, 598, 728, 875, 1040, 1224, 1428, 1653, 1900, 2170, 2464, 2783, 3128, 3500, 3900, 4329, 4788, 5278, 5800, 6355, 6944, 7568, 8228, 8925, 9660, 10434, 11248, 12103, 13000, 13940, 14924, 15953, 17028, 18150, 19320
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OFFSET
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3,1
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COMMENTS
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A covering of a set is a tricovering if every element of the set is covered by exactly three blocks of the covering.
If Y is a 4-subset of an n-set X then, for n>=6, a(n-3) is the number of 3-subsets of X having at most one element in common with Y. - Milan Janjic, Dec 08 2007
Also the number of balls in a triangular pyramid of which all balls located on the edges have been removed such that the remaining pyramid's edges each consist of two adjacent balls. The layers of pyramids of this form start (from the top) 3, 7, 12, 18, 25, 33,... (A055998) with one smaller additional layer 1, 3, 6, 10, 15, 21,... (A000217) at the bottom. Thus, a(n) = A000217(n) + Sum_{k=1..n} A055998(k). Example: a(4) = (3+7+12+18)+10 = 50. - K. G. Stier, Dec 12 2012
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LINKS
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FORMULA
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a(n) = binomial(n+3, 3) - 6*binomial(n+1, 1) + 8*binomial(n, 0) - 3*binomial(n-1, -1).
G.f.: -y^3*(-4+3*y)/(-1+y)^4.
E.g.f. for ordered k-block tricoverings of an unlabeled n-set is exp(-x+x^2/2+x^3/3*y/(1-y)) * sum(k>=0, 1/(1-y)^binomial(k, 3)*exp(-x^2/2*1/(1-y)^n)*x^k/k! ).
a(n) = (n+9)*binomial(n-1, 2)/3.
a(n) = (n-2)*(n-1)*(n+9)/6. - Zak Seidov, Jun 15 2006
a(3)=4, a(4)=13, a(5)=28, a(6)=50, a(n) = 4*a(n-1)-6*a(n-2)+ 4*a(n-3)- a(n-4). - Harvey P. Dale, Jul 21 2012
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MATHEMATICA
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Table[(n-2)(n-1)(n+9)/6, {n, 3, 50}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {4, 13, 28, 50}, 50] (* Harvey P. Dale, Jul 21 2012 *)
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PROG
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CROSSREFS
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Fourth column (m=3) of (1, 4)-Pascal triangle A095666.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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