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A366542
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Number of discrete uninorms defined on the finite chain L_n={0,1,...n}, U:L_n^2->L_n, whose underlying operators are smooth and idempotent, or smooth and idempotent-free.
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0
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2, 6, 14, 30, 56, 100, 178, 322, 596, 1128, 2174, 4246, 8368, 16588, 33002, 65802, 131372, 262480, 524662, 1048990, 2097608, 4194804, 8389154, 16777810, 33555076, 67109560, 134218478, 268436262, 536871776, 1073742748, 2147484634, 4294968346, 8589935708, 17179870368, 34359739622
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OFFSET
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1,1
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COMMENTS
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The number of discrete uninorms defined on the finite chain L_n={0,1,...n} whose underlying operators are smooth and idempotent or smooth and idempotent-free, i.e., the number of monotonic increasing binary functions U:L_n^2->L_n such that U is associative (U(x,U(y,z))=U(U(x,y),z) for all x,y,z in L_N), U is commutative (U(x,y)=U(y,x) for all x,y in L_n) and has some neutral element e in L_n (U(x,e)=U(e,x)=x for all x in L_n), such that U restricted to {0,...,e} and to {e,...,n} is smooth and idempotent, or smooth and idempotent-free.
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LINKS
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FORMULA
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a(1)=2, a(2)=6 and a(n) = 2^n+n*(n+1) - 6 for n>=3.
G.f.: 2*x*(1 - 2*x + x^2 - 3x^4 + 2*x^5)/((1 - x)^3*(1 - 2*x)).
a(n) = A131924(n) - 6 for n>=3. (End)
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MATHEMATICA
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Join[{1, 6}, Table[2^n + n + n^2 - 6, {n, 3, 35}]]
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CROSSREFS
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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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