This site is supported by donations to The OEIS Foundation.

 Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing. Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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 Études 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:=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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

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