OFFSET
0,3
COMMENTS
Number of (s(0), s(1), ..., s(n)) such that 0 < s(i) < 6 and |s(i) - s(i-1)| <= 1 for i = 1,2,...,n, s(0) = 3, s(n) = 3.
In general, a(n,m,j,k) = (2/m)*Sum_{r=1..m-1} sin(j*r*Pi/m)*sin(k*r*Pi/m)*(1+2*cos(Pi*r/m))^n is the number of (s(0), s(1), ..., s(n)) such that 0 < s(i) < m and |s(i) - s(i-1)| <= 1 for i = 1,2,...,n, s(0) = j, s(n) = k. - Herbert Kociemba, Jun 02 2004
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.
Denis Chebikin and Richard Ehrenborg, The f-vector of the descent polytope, arXiv:0812.1249 [math.CO], 2008-2010; Disc. Comput. Geom., 45 (2011), 410-424.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1007
Alina F. Y. Zhao, Bijective proofs for some results on the descent polytope, Australasian Journal of Combinatorics, Volume 65(1) (2016), Pages 45-52.
Index entries for linear recurrences with constant coefficients, signature (3,0,-2).
FORMULA
a(n) = 2*a(n-1) + 2*a(n-2) - 1.
a(n) = Sum_{alpha=RootOf(1-3*z+2*z^3)} alpha^(-n)/3.
a(n) = (1 + (1+sqrt(3))^n + (1-sqrt(3))^n)/3. Binomial transform of A025192 (with interpolated zeros). - Paul Barry, Sep 16 2003
a(n) = (1/3)*Sum_{k=1..5} sin(Pi*k/2)^2 * (1 + 2*cos(Pi*k/6))^n. - Herbert Kociemba, Jun 02 2004
a(0)=1, a(1)=1, a(2)=3, a(n) = 3*a(n-1) - 2*a(n-3). - Harvey P. Dale, Aug 22 2012
E.g.f.: exp(x)*(1 + 2*cosh(sqrt(3)*x))/3. - Stefano Spezia, Mar 02 2024
MAPLE
spec := [S, {S=Sequence(Prod(Union(Sequence(Prod(Sequence(Z), Z)), Z), Z))}, unlabeled ]: seq(combstruct[count ](spec, size=n), n=0..20);
seq(coeff(series((1-2*x)/(1-3*x+2*x^3), x, n+1), x, n), n = 0 .. 40); # G. C. Greubel, Oct 21 2019
MATHEMATICA
CoefficientList[Series[(1-2x)/(1-3x+2x^3), {x, 0, 30}], x] (* or *) LinearRecurrence[{3, 0, -2}, {1, 1, 3}, 30] (* Harvey P. Dale, Aug 22 2012 *)
PROG
(Sage) from sage.combinat.sloane_functions import recur_gen2b; it = recur_gen2b(1, 1, 2, 2, lambda n: -1); [next(it) for i in range(0, 29)] # Zerinvary Lajos, Jul 09 2008
(PARI) Vec((1-2*x)/(1-3*x+2*x^3)+O(x^30))
(Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-2*x)/(1-3*x+2*x^3) )); // G. C. Greubel, Oct 21 2019
(GAP) a:=[1, 1, 3];; for n in [4..30] do a[n]:=3*a[n-1]-2*a[n-3]; od; a; # G. C. Greubel, Oct 21 2019
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
encyclopedia(AT)pommard.inria.fr, Jan 25 2000
EXTENSIONS
More terms from James A. Sellers, Jun 06 2000
Definition revised by N. J. A. Sloane, Feb 24 2011
STATUS
approved