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A052948 G.f.: (1-2*x)/(1-3*x+2*x^3) 4
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

Table of n, a(n) for n=0..27.

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

Recurrence: {a(1)=1, a(0)=1, -2*a(n)-2*a(n+1)+a(n+2)+1}

Sum(1/3*_alpha^(-n), _alpha=RootOf(1-3*_Z+2*_Z^3))

a(n)=1/3+(1+sqrt(3))^n/3+(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+2Cos(Pi*k/6))^n) - Herbert Kociemba, Jun 02 2004

a(0) = a(1) = 1; a(n+2) = 2*a(n+1) + 2*a(n) - 1. - Carl R. White, Jul 30 2009

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

MAPLE

spec := [S, {S=Sequence(Prod(Union(Sequence(Prod(Sequence(Z), Z)), Z), Z))}, unlabeled ]: seq(combstruct[count ](spec, size=n), n=0..20);

MATHEMATICA

Join[{a=1, b=1}, Table[c=(a+b)*2-1; a=b; b=c, {n, 0, 60}]] (* Vladimir Joseph Stephan Orlovsky, Nov 22 2010 *)

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 xrange(0, 29)] # Zerinvary Lajos, Jul 09 2008

(PARI) Vec((1-2*x)/(1-3*x+2*x^3)+O(x^99))

CROSSREFS

Cf. A026150.

Sequence in context: A087224 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

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Last modified October 20 15:44 EDT 2018. Contains 316389 sequences. (Running on oeis4.)