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

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

• easy
• nonn

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
• Yassine Otmani, The 2-Pascal Triangle and a Related Riordan Array, J. Int. Seq. (2025) Vol. 28, Issue 3, Art. No. 25.3.5. See p. 12.
• 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).

Crossrefs

Cf. A026150, A077846.

Programs

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

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-2*x)/(1-3*x+2*x^3) )); // G. C. Greubel, Oct 21 2019

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 *)

PARI

Vec((1-2*x)/(1-3*x+2*x^3)+O(x^30))

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

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

Extensions

More terms from James Sellers, Jun 06 2000

Definition revised by N. J. A. Sloane, Feb 24 2011