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

0,3

Comments

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

Links

Table of n, a(n) for n=0..33.

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).

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)

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 *)

Prog

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

Crossrefs

Sequence in context: A265054 A099018 A033484 * A266373 A266374 A008267

Adjacent sequences: A296950 A296951 A296952 * A296954 A296955 A296956

Keyword

nonn,easy

Author

J. Devillet, Dec 22 2017

Extensions

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

Status

approved