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%I #38 Sep 08 2022 08:44:59
%S 2,1,3,4,9,16,33,64,129,256,513,1024,2049,4096,8193,16384,32769,65536,
%T 131073,262144,524289,1048576,2097153,4194304,8388609,16777216,
%U 33554433,67108864,134217729,268435456,536870913,1073741824,2147483649
%N Expansion of (2-3*x-x^2+x^3)/((1-x)*(1+x)*(1-2*x)).
%C Equals row sums of triangle A178616 but replacing the 2 with a 1. - _Gary W. Adamson_, May 30 2010
%C Inverse binomial transform is (-1)^n * a(n). - _Michael Somos_, Jun 03 2014
%C An autosequence of the second kind whose first kind companion is A005578. - _Jean-François Alcover_, Mar 18 2020
%H G. C. Greubel, <a href="/A052950/b052950.txt">Table of n, a(n) for n = 0..1000</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=1009">Encyclopedia of Combinatorial Structures 1009</a>
%H OEIS Wiki, <a href="https://oeis.org/wiki/Autosequence">Autosequence</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-2).
%F G.f.: (2-3*x-x^2+x^3)/((1-x^2)*(1-2*x)).
%F a(n) = a(n-1) + 2*a(n-2) - 1.
%F a(n) = 2^(n-1) + Sum_{alpha=RootOf(-1+z^2)} alpha^(-n)/2.
%F From _Paul Barry_, Sep 18 2003: (Start)
%F a(n) = (2^n + 1 + (-1)^n + 0^n)/2.
%F E.g.f.: cosh(x)*(1+exp(x)). (End)
%F a(2*n + 1) = 4 * a(2*n - 1) for all n in Z. a(2*n + 2) = 3*a(2*n + 1) + 2*a(2*n) if n>0. - _Michael Somos_, Jun 04 2014
%e G.f. = 2 + x + 3*x^2 + 4*x^3 + 9*x^4 + 16*x^5 + 33*x^6 + 64*x^7 + 129*x^8 + ...
%p spec:= [S,{S=Union(Sequence(Prod(Sequence(Z),Z)), Sequence(Prod(Z,Z)))}, unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);
%p seq(`if`(n=0, 2, (2^n +1 +(-1)^n)/2), n=0..40); # _G. C. Greubel_, Oct 21 2019
%t a[n_]:= (2^n +1 +(-1)^n +Boole[n==0])/2; (* _Michael Somos_, Jun 03 2014 *)
%t a[n_]:= If[ n<0, (1-n)! SeriesCoefficient[Sinh[x] +Exp[x/2], {x,0,1-n}], n! SeriesCoefficient[Cosh[x](1+Exp[x]), {x,0,n}]]; (* _Michael Somos_, Jun 04 2014 *)
%t LinearRecurrence[{2,1,-2}, {2,1,3,4}, 40] (* _G. C. Greubel_, Oct 21 2019 *)
%o (PARI) {a(n)=(2^n+1+(-1)^n+(n==0))/2}; /* _Michael Somos_, Jun 03 2014 */
%o (Magma) [2] cat [(2^n +1 +(-1)^n)/2: n in [1..40]]; // _G. C. Greubel_, Oct 21 2019
%o (Sage) [2]+[(2^n +1 +(-1)^n)/2 for n in (1..40)] # _G. C. Greubel_, Oct 21 2019
%o (GAP) Concatenation([2], List([1..40], n-> (2^n +1 +(-1)^n)/2)); # _G. C. Greubel_, Oct 21 2019
%Y Cf. A178616. - _Gary W. Adamson_, May 30 2010
%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
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