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A366542 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.

%I A366542 #16 Nov 06 2023 14:06:21
%S A366542 2,6,14,30,56,100,178,322,596,1128,2174,4246,8368,16588,33002,65802,
%T A366542 131372,262480,524662,1048990,2097608,4194804,8389154,16777810,
%U A366542 33555076,67109560,134218478,268436262,536871776,1073742748,2147484634,4294968346,8589935708,17179870368,34359739622
%N A366542 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.
%C A366542 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.
%H A366542 D. Ruiz-Aguilera and J. Torrens, <a href="https://doi.org/10.1016/j.fss.2014.10.020">A characterization of discrete uninorms having smooth underlying operators</a>, Fuzzy Sets and Systems, Volume 268, 2015, 44-58.
%H A366542 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (5,-9,7,-2).
%F A366542 a(1)=2, a(2)=6 and a(n) = 2^n+n*(n+1) - 6 for n>=3.
%F A366542 From _Stefano Spezia_, Nov 05 2023: (Start)
%F A366542 G.f.: 2*x*(1 - 2*x + x^2 - 3x^4 + 2*x^5)/((1 - x)^3*(1 - 2*x)).
%F A366542 a(n) = A131924(n) - 6 for n>=3. (End)
%t A366542 Join[{1, 6}, Table[2^n + n + n^2 - 6, {n, 3, 35}]]
%Y A366542 Cf. A131924.
%K A366542 nonn,easy
%O A366542 1,1
%A A366542 _Marc Munar_, Oct 12 2023