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A083667
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Number of antisymmetric binary relations on a set of n labeled points.
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9
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1, 2, 12, 216, 11664, 1889568, 918330048, 1338925209984, 5856458868470016, 76848453272063549952, 3025216211508053707410432, 357271984146678126737757198336, 126579320351263180234426948827254784
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OFFSET
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0,2
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COMMENTS
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The number of infinite sets per level n in the Collatz Tree partitioning the inverse iterates of the number m, where the value n is given by the number of odds in sequences which converge to m in the 3x+1 Problem, and m is a positive odd integer, not divisible by 3, for which the 3x+1 Problem holds. We get the entire Collatz (3x+1) Tree for the case m = 1. Otherwise, we get a portion of the tree. We can think of the infinite sets as residue classes modulo powers of 3. In this way Wirsching gets an abstract description of the Collatz Tree along with its underlying combinatorial structure. See Corollary 3.2 in Wirsching's paper. - Jeffrey R. Goodwin, Jul 26 2011
Let T_n denote the n X n matrix with T_n(i,j) = 3^(min(i,j)-1); then a(n) = det(T_(n+1)). - Lechoslaw Ratajczak, May 11 2021
The number of simple directed graphs (digraphs) on n labeled vertices, allowing for loops but restricting each pair of distinct vertices to have at most one directed edge between them, in either direction. These constraints define a structure where each vertex can have a loop, and for any two distinct vertices, there are three possible relationships: no edge, a directed edge from the first vertex to the second, or a directed edge from the second vertex to the first. - Constantinos Kourouzides, Mar 25 2024
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LINKS
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FORMULA
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a(n) = 2^n * 3^((n*(n-1))/2).
2*Sum_{n>=2) 1/a(n) = 2*Sum_{n>=2} 2^(-n)*3^(-((n*(n-1))/2)) = Sum_{n>=1} 1/Product_{k=1..n} A008776(k) = Sum_{n>=1} 1/Product_{k=1..n} 2*3^k = 0.1760984543123346169209966002213.... - Alexander R. Povolotsky, Aug 08 2011
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MAPLE
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MATHEMATICA
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PROG
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(GAP) a := n -> 2^n * 3^Binomial(n, 2);
(PARI) a(n)=2^n*3^((n^2-n)/2)
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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Goetz Pfeiffer (Goetz.Pfeiffer(AT)nuigalway.ie), May 02 2003
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EXTENSIONS
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STATUS
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approved
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