|
%I #61 Mar 02 2024 12:34:25
%S 1,1,3,7,19,51,139,379,1035,2827,7723,21099,57643,157483,430251,
%T 1175467,3211435,8773803,23970475,65488555,178918059,488813227,
%U 1335462571,3648551595,9968028331,27233159851,74402376363,203271072427,555346897579,1517235940011,4145165675179
%N Expansion of g.f.: (1-2*x)/(1-3*x+2*x^3).
%C 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.
%C 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
%H G. C. Greubel, <a href="/A052948/b052948.txt">Table of n, a(n) for n = 0..1000</a>
%H Paul Barry, <a href="https://arxiv.org/abs/1803.06408">Three Études on a sequence transformation pipeline</a>, arXiv:1803.06408 [math.CO], 2018.
%H Denis Chebikin and Richard Ehrenborg, <a href="https://arxiv.org/abs/0812.1249">The f-vector of the descent polytope</a>, arXiv:0812.1249 [math.CO], 2008-2010; Disc. Comput. Geom., 45 (2011), 410-424.
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=1007">Encyclopedia of Combinatorial Structures 1007</a>
%H Alina F. Y. Zhao, <a href="https://ajc.maths.uq.edu.au/pdf/65/ajc_v65_p045.pdf">Bijective proofs for some results on the descent polytope</a>, Australasian Journal of Combinatorics, Volume 65(1) (2016), Pages 45-52.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,0,-2).
%F a(n) = 2*a(n-1) + 2*a(n-2) - 1.
%F a(n) = Sum_{alpha=RootOf(1-3*z+2*z^3)} alpha^(-n)/3.
%F a(n) = (1 + (1+sqrt(3))^n + (1-sqrt(3))^n)/3. Binomial transform of A025192 (with interpolated zeros). - _Paul Barry_, Sep 16 2003
%F 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
%F 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
%F a(n) = A077846(n) - 2*A077846(n-1). - _R. J. Mathar_, Feb 27 2019
%F E.g.f.: exp(x)*(1 + 2*cosh(sqrt(3)*x))/3. - _Stefano Spezia_, Mar 02 2024
%p spec := [S,{S=Sequence(Prod(Union(Sequence(Prod(Sequence(Z),Z)),Z),Z))}, unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);
%p 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
%t 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 *)
%o (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
%o (PARI) Vec((1-2*x)/(1-3*x+2*x^3)+O(x^30))
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-2*x)/(1-3*x+2*x^3) )); // _G. C. Greubel_, Oct 21 2019
%o (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
%Y Cf. A026150, A077846.
%K easy,nonn
%O 0,3
%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E More terms from _James A. Sellers_, Jun 06 2000
%E Definition revised by _N. J. A. Sloane_, Feb 24 2011
|
