A052948 Expansion of g.f.: (1-2*x)/(1-3*x+2*x^3).

1, 1, 3, 7, 19, 51, 139, 379, 1035, 2827, 7723, 21099, 57643, 157483, 430251, 1175467, 3211435, 8773803, 23970475, 65488555, 178918059, 488813227, 1335462571, 3648551595, 9968028331, 27233159851, 74402376363, 203271072427, 555346897579, 1517235940011, 4145165675179

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

a(n) = A077846(n) - 2*A077846(n-1). - R. J. Mathar, Feb 27 2019

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

Cf. A026150, A077846.

Sequence in context: A341703 A078059 A018031 * A026325 A002426 A011769

Adjacent sequences: A052945 A052946 A052947 * A052949 A052950 A052951

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