A296953 Number of bisymmetric, quasitrivial, and order-preserving binary operations on the n-element set {1,...,n}.

0, 1, 4, 10, 22, 46, 94, 190, 382, 766, 1534, 3070, 6142, 12286, 24574, 49150, 98302, 196606, 393214, 786430, 1572862, 3145726, 6291454, 12582910, 25165822, 50331646, 100663294, 201326590, 402653182, 805306366, 1610612734, 3221225470, 6442450942, 12884901886

Offset: 0

Author

J. Devillet, Dec 22 2017

Keywords

• nonn
• easy

Comments

Apart from the offset the same as A033484. - R. J. Mathar, Alois P. Heinz, Jan 02 2018

Links

• J. Devillet, Bisymmetric and quasitrivial operations: characterizations and enumerations, arXiv:1712.07856 [math.RA] (2017).
• Index entries for linear recurrences with constant coefficients, signature (3,-2).

Programs

Mathematica

Nest[Append[#, 2 Last@ # + 2] &, {0, 1}, 32] (* or *)
Array[3*2^(# - 1) - 2 + Boole[# == 0]/2 &, 34, 0] (* or *)
CoefficientList[Series[x (1 + x)/((1 - x) (1 - 2 x)), {x, 0, 33}], x] (* Michael De Vlieger, Dec 22 2017 *)

PARI

concat(0, Vec(x*(1 + x) / ((1 - x)*(1 - 2*x)) + O(x^40))) \\ Colin Barker, Dec 22 2017

Formula

a(0)=0, a(1)=1, a(n+1)-2*a(n) = 2.

From Colin Barker, Dec 22 2017: (Start)

G.f.: x*(1 + x) / ((1 - x)*(1 - 2*x)).

a(n) = 3*2^(n-1) - 2 for n>0.

a(n) = 3*a(n-1) - 2*a(n-2) for n>2.

(End)

Extensions

G.f. replaced by a better g.f. by Colin Barker, Dec 23 2017