|
%I #43 Sep 08 2022 08:44:59
%S 2,2,4,6,12,22,44,86,172,342,684,1366,2732,5462,10924,21846,43692,
%T 87382,174764,349526,699052,1398102,2796204,5592406,11184812,22369622,
%U 44739244,89478486,178956972,357913942,715827884,1431655766,2863311532
%N Expansion of 2*(1-x-x^2)/((1-x)*(1+x)*(1-2*x)).
%C a(n) = sum of absolute values of terms in the (n+1)-th row of the triangle in A108561; - _Reinhard Zumkeller_, Jun 10 2005
%C a(n) = A078008(n+1) + 2*(1 + n mod 2). - _Reinhard Zumkeller_, Jun 10 2005
%C Essentially the same as A128209. - _R. J. Mathar_, Jun 14 2008
%H G. C. Greubel, <a href="/A052953/b052953.txt">Table of n, a(n) for n = 0..1000</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=1024">Encyclopedia of Combinatorial Structures 1024</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-2).
%F G.f.: 2*(1-x-x^2)/((1-x^2)*(1-2*x)).
%F a(n) = a(n-1) + 2*a(n-2) - 2.
%F a(n) = 1 + Sum_{alpha=RootOf(-1+z+2*z^2)} (1 + 4*alpha)*alpha^(-1-n)/9.
%F a(2n) = 2*a(n-1)-2, a(2n+1) = 2*a(2n). - _Lee Hae-hwang_, Oct 11 2002
%F From _Paul Barry_, May 24 2004: (Start)
%F a(n) = A001045(n+1) + 1.
%F a(n) = (2^(n+1) - (-1)^(n+1))/3 + 1. (End)
%F E.g.f.: (2*exp(2*x) + 3*exp(x) + exp(-x))/3. - _G. C. Greubel_, Oct 21 2019
%p spec:= [S,{S=Union(Sequence(Union(Prod(Union(Z,Z),Z),Z)),Sequence(Z))}, unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);
%p seq((2^(n+1) +3 +(-1)^n)/3, n=0..40); # _G. C. Greubel_, Oct 21 2019
%t LinearRecurrence[{2,1,-2}, {2,2,4}, 40] (* _G. C. Greubel_, Oct 22 2019 *)
%o (Sage) [(2^(n+1) +3 +(-1)^n)/3 for n in (0..40)] # _G. C. Greubel_, Oct 21 2019
%o (PARI) vector(41, n, (2^n +3 -(-1)^n)/3 ) \\ _G. C. Greubel_, Oct 21 2019
%o (Magma) [(2^(n+1) +3 +(-1)^n)/3: n in [0..40]]; // _G. C. Greubel_, Oct 21 2019
%o (GAP) List([0..40], n-> (2^(n+1) +3 +(-1)^n)/3); # _G. C. Greubel_, Oct 21 2019
%Y Apart from initial term, equals A026644(n+1) + 2.
%Y Cf. A001045.
%K easy,nonn
%O 0,1
%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E More terms from _James A. Sellers_, Jun 05 2000
|
