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A052948 Expansion of g.f.: (1-2*x)/(1-3*x+2*x^3). 5
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 (list; graph; refs; listen; history; text; internal format)
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+2Cos(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

REFERENCES

Denis Chebikin and Richard Ehrenborg, The f-vector of the descent polytope, Disc. Comput. Geom., 45 (2011), 410-424.

Alina F. Y. Zhao, Bijective proofs for some results on the descent polytope, AUSTRALASIAN JOURNAL OF COMBINATORICS, Volume 65(1) (2016), Pages 45-52.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

Paul Barry, Three √Čtudes on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1007

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

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); [it.next() 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.

Sequence in context: A308398 A078059 A018031 * A026325 A002426 A011769

Adjacent sequences:  A052945 A052946 A052947 * A052949 A052950 A052951

KEYWORD

easy,nonn,changed

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

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Last modified December 16 04:05 EST 2019. Contains 330013 sequences. (Running on oeis4.)