OFFSET
0,3
COMMENTS
Number of 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 weak ordering R.
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
Andrei Asinowski, Cyril Banderier, Sara Billey, Benjamin Hackl, Svante Linusson, Pop-stack sorting and its image: permutations with overlapping runs, Eurocomb, Acta Math. Univ. Comenianae (2019), 1-8.
M. Couceiro, J. Devillet, and J.-L. Marichal, Quasitrivial semigroups: characterizations and enumerations, arXiv:1709.09162 [math.RA], 2017.
Index entries for linear recurrences with constant coefficients, signature (3,-1,-1).
FORMULA
G.f.: 2*x^2 / (x^3 + x^2 - 3x + 1).
a(n) = 2*A048739(n-2), a(0) = a(1) = 0.
From Colin Barker, Sep 28 2017: (Start)
a(n) = 3*a(n-1) - a(n-2) - a(n-3) for n > 2.
a(n) = (-2 + (1-sqrt(2))^n + (1+sqrt(2))^n) / 2. (End)
a(n) = A265278(n) for n != 1. - Joerg Arndt, Oct 01 2017
MAPLE
A293004:=gfun:-rectoproc({a(n)=3*a(n-1) -a(n-2)-a(n-3), a(0)=0, a(1)=0, a(2)=2}, a(n), remember): map(A293004, [$0..10^3]); # Muniru A Asiru, Oct 09 2017
MATHEMATICA
CoefficientList[Series[2 x^2/(x^3 + x^2 - 3 x + 1), {x, 0, 30}], x] (* Michael De Vlieger, Oct 06 2017 *)
RecurrenceTable[{a[1]==a[2]==0, a[3]==2, a[n]==3a[n-1] - a[n-2] - a[n-3]}, a, {n, 40}] (* Vincenzo Librandi, Oct 09 2017 *)
PROG
(PARI) concat(vector(2), Vec(2*x^2 / (x^3+x^2-3*x+1) + O(x^40))) \\ Colin Barker, Sep 28 2017
(Magma) I:=[0, 0, 2]; [n le 3 select I[n] else 3*Self(n-1)-Self(n-2)-Self(n-3): n in [1..40]]; // Vincenzo Librandi, Oct 09 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
J. Devillet, Sep 28 2017
STATUS
approved