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.

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

Offset: 1

Author

Marc Munar, Oct 12 2023

Keywords

• nonn
• easy

Comments

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.

Links

• D. Ruiz-Aguilera and J. Torrens, A characterization of discrete uninorms having smooth underlying operators, Fuzzy Sets and Systems, Volume 268, 2015, 44-58.
• Index entries for linear recurrences with constant coefficients, signature (5,-9,7,-2).

Crossrefs

Cf. A131924.

Programs

Mathematica

Join[{1, 6}, Table[2^n + n + n^2 - 6, {n, 3, 35}]]

Formula

a(1)=2, a(2)=6 and a(n) = 2^n+n*(n+1) - 6 for n>=3.

From Stefano Spezia, Nov 05 2023: (Start)

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)