A296953 - OEIS

%I #20 Dec 22 2018 16:10:28

%S 0,1,4,10,22,46,94,190,382,766,1534,3070,6142,12286,24574,49150,98302,

%T 196606,393214,786430,1572862,3145726,6291454,12582910,25165822,

%U 50331646,100663294,201326590,402653182,805306366,1610612734,3221225470,6442450942,12884901886

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

%C Apart from the offset the same as A033484. - _R. J. Mathar_, _Alois P. Heinz_, Jan 02 2018

%H J. Devillet, <a href="https://arxiv.org/abs/1712.07856">Bisymmetric and quasitrivial operations: characterizations and enumerations</a>,  arXiv:1712.07856 [math.RA] (2017).

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2).

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

%F From _Colin Barker_, Dec 22 2017: (Start)

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

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

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

%F (End)

%t Nest[Append[#, 2 Last@ # + 2] &, {0, 1}, 32] (* or *)

%t Array[3*2^(# - 1) - 2 + Boole[# == 0]/2 &, 34, 0] (* or *)

%t CoefficientList[Series[x (1 + x)/((1 - x) (1 - 2 x)), {x, 0, 33}], x] (* _Michael De Vlieger_, Dec 22 2017 *)

%o (PARI) concat(0, Vec(x*(1 + x) / ((1 - x)*(1 - 2*x)) + O(x^40))) \\ _Colin Barker_, Dec 22 2017

%K nonn,easy

%O 0,3

%A _J. Devillet_, Dec 22 2017

%E G.f. replaced by a better g.f. by _Colin Barker_, Dec 23 2017