OFFSET
0,3
COMMENTS
a(n) = A000225(n)-1, a(0)=0, a(1)=1. Number of quasilinear weak orderings R on {1,...,n} that are weakly single-peaked w.r.t. the total ordering 1<...<n and for which {1,...,n} has exactly one maximal element for the quasilinear weak ordering R.
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
J. Devillet, Bisymmetric and quasitrivial operations: characterizations and enumerations, arXiv:1712.07856 [math.RA], 2017-2018.
Index entries for linear recurrences with constant coefficients, signature (3,-2).
FORMULA
From Colin Barker, Dec 22 2017: (Start)
G.f.: x*(1 - x + 2*x^2) / ((1 - x)*(1 - 2*x)).
a(n) = 2^n - 2 for n>1.
a(n) = 3*a(n-1) - 2*a(n-2) for n>3. (End)
a(n) = A134067(n-2) for n >= 3. - Georg Fischer, Oct 30 2018
E.g.f.: 1 + exp(x)*(exp(x) - 2) + x. - Stefano Spezia, May 07 2023
MATHEMATICA
CoefficientList[Series[x (1 - x + 2 x^2)/((1 - x) (1 - 2 x)), {x, 0, 33}], x] (* or *)
LinearRecurrence[{3, -2}, {0, 1, 2, 6}, 34] (* Michael De Vlieger, Dec 22 2017 *)
PROG
(PARI) concat(0, Vec(x*(1 - x + 2*x^2) / ((1 - x)*(1 - 2*x)) + O(x^40))) \\ Colin Barker, Dec 22 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
J. Devillet, Dec 22 2017
STATUS
approved