A052951 - OEIS

%I #64 Oct 01 2022 06:09:33

%S 1,5,14,36,88,208,480,1088,2432,5376,11776,25600,55296,118784,253952,

%T 540672,1146880,2424832,5111808,10747904,22544384,47185920,98566144,

%U 205520896,427819008,889192448,1845493760,3825205248,7918845952

%N Expansion of (1 + x - 2*x^2)/(1 - 2*x)^2.

%C Equals binomial transform of A042948 starting with "1": (1, 4, 5, 8, 9, 12, 13, ...) = terms > 0, == 0 or 1 mod 4. - _Gary W. Adamson_, Feb 07 2009

%H Vincenzo Librandi, <a href="/A052951/b052951.txt">Table of n, a(n) for n = 0..1000</a>

%H O. Aichholzer, A. Asinowski, and T. Miltzow, <a href="http://arxiv.org/abs/1403.5546">Disjoint compatibility graph of non-crossing matchings of points in convex position</a>, arXiv preprint arXiv:1403.5546 [math.CO], 2014.

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=1021">Encyclopedia of Combinatorial Structures 1021</a>.

%H Agustín Moreno Cañadas, Hernán Giraldo, Gabriel Bravo Rios, <a href="http://dx.doi.org/10.17654/MS101081631">On the Number of Sections in the Auslander-Reiten Quiver of Algebras of Dynkin Type</a>, Far East Journal of Mathematical Sciences (FJMS), Vol. 101, No. 8 (2017), pp. 1631-1654.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-4).

%F G.f.: (1+x-2*x^2)/(1-2*x)^2.

%F a(n) = 4*(a(n-1) - a(n-2)).

%F a(n) = (n+1)*2^n + 2^(n-1), n > 0.

%F a(n) = A118413(n+1,n-1) for n > 2. - _Reinhard Zumkeller_, Apr 27 2006

%F E.g.f.: (1/2)*(-1 + exp(2*x)*(3 + 4*x)). - _Stefano Spezia_, Oct 22 2019

%F From _Amiram Eldar_, Oct 01 2022: (Start)

%F Sum_{n>=0} 1/a(n) = 4*sqrt(2)*arcsinh(1) - 11/3.

%F Sum_{n>=0} (-1)^n/a(n) = 13/3 - 4*sqrt(2)*arccot(sqrt(2)). (End)

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

%p seq(`if`(n=0, 1, 2^(n-1)*(2*n+3)), n=0..40); # _G. C. Greubel_, Oct 21 2019

%t CoefficientList[Series[(1+x-2*x^2)/(1-2*x)^2,{x,0,40}],x] (* _Vincenzo Librandi_, Jun 22 2012 *)

%t LinearRecurrence[{4,-4}, {1,5,14}, 40] (* _G. C. Greubel_, Oct 21 2019 *)

%o (Magma) I:=[1, 5, 14]; [n le 3 select I[n] else 4*Self(n-1)-4*Self(n-2): n in [1..40]]; // _Vincenzo Librandi_, Jun 22 2012

%o (PARI) x='x+O('x^40); Vec((1+x-2*x^2)/(1-2*x)^2) \\ _Altug Alkan_, Mar 03 2018

%o (Sage) [1]+[2^(n-1)*(2*n+3) for n in (1..40)] # _G. C. Greubel_, Oct 21 2019

%o (GAP) Concatenation([1], List([1..40], n-> 2^(n-1)*(2*n+3) )); # _G. C. Greubel_, Oct 21 2019

%Y Cf. A042948, A118413.

%K easy,nonn

%O 0,2

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