A052957 Expansion of 2*(1-x-x^2)/((1-2*x)*(1-2*x^2)).

2, 2, 6, 8, 20, 32, 72, 128, 272, 512, 1056, 2048, 4160, 8192, 16512, 32768, 65792, 131072, 262656, 524288, 1049600, 2097152, 4196352, 8388608, 16781312, 33554432, 67117056, 134217728, 268451840, 536870912, 1073774592, 2147483648

Offset: 0

Author

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

Keywords

• easy
• nonn

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1028
• Index entries for linear recurrences with constant coefficients, signature (2,2,-4).

Programs

GAP

a:=[2,2,6];; for n in [4..30] do a[n]:=2*a[n-1]+2*a[n-2]-4*a[n-3]; od; a; # G. C. Greubel, Oct 22 2019

Magma

[2] cat [Round(2^n +2^((n-1)/2)*(1+(-1)^n)/Sqrt(2)): n in [1..30]]; // G. C. Greubel, Oct 22 2019

Maple

spec:= [S,{S=Union(Sequence(Prod(Union(Z,Z),Z)),Sequence(Union(Z,Z)))}, unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);
seq(coeff(series(2*(1-x-x^2)/((1-2*x)*(1-2*x^2)), x, n+1), x, n), n = 0 .. 40); # G. C. Greubel, Oct 22 2019

Mathematica

CoefficientList[Series[2*(1-x-x^2)/((1-2*x)*(1-2*x^2)), {x, 0, 31}], x] (* Michael De Vlieger, Sep 23 2016 *)
Join[{2}, Table[2^n +2^((n-1)/2)*(1+(-1)^n)/Sqrt[2], {n, 30}]] (* G. C. Greubel, Oct 22 2019 *)
LinearRecurrence[{2,2,-4},{2,2,6},40] (* Harvey P. Dale, Jul 19 2020 *)

PARI

a(n)=2^n+if(n%2,,2^(n/2)) \\ Charles R Greathouse IV, Sep 23 2016

Sage

[2]+[2^n +2^((n-1)/2)*(1+(-1)^n)/sqrt(2) for n in (1..30)] # G. C. Greubel, Oct 22 2019

Formula

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

a(n) = 2*a(n-1) + 2*a(n-2) - 4*a(n-3).

a(n) = 2^n + Sum_{alpha=RootOf(-1+2*x^2)} alpha^(-n)/2.

a(n) = 2*A051437(n+1), n > 0. - R. J. Mathar, Nov 27 2011

From Colin Barker, Sep 23 2016: (Start)

a(n) = 2^(n/2) + 2^n for n even.

a(n) = 2^n for n odd.

(End)

E.g.f.: (1/2)*(2*exp(2*x) + exp(-sqrt(2)*x) + exp(sqrt(2)*x)). - Stefano Spezia, Oct 22 2019

Extensions

More terms from James Sellers, Jun 05 2000