A052950 Expansion of (2-3*x-x^2+x^3)/((1-x)*(1+x)*(1-2*x)).

2, 1, 3, 4, 9, 16, 33, 64, 129, 256, 513, 1024, 2049, 4096, 8193, 16384, 32769, 65536, 131073, 262144, 524289, 1048576, 2097153, 4194304, 8388609, 16777216, 33554433, 67108864, 134217729, 268435456, 536870913, 1073741824, 2147483649

Offset: 0

Author

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

Keywords

• easy
• nonn

Comments

Equals row sums of triangle A178616 but replacing the 2 with a 1. - Gary W. Adamson, May 30 2010

Inverse binomial transform is (-1)^n * a(n). - Michael Somos, Jun 03 2014

An autosequence of the second kind whose first kind companion is A005578. - Jean-François Alcover, Mar 18 2020

Examples

			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 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1009
• OEIS Wiki, Autosequence
• Index entries for linear recurrences with constant coefficients, signature (2,1,-2).

Crossrefs

Cf. A178616. - Gary W. Adamson, May 30 2010

Programs

GAP

Concatenation([2], List([1..40], n-> (2^n +1 +(-1)^n)/2));  # G. C. Greubel, Oct 21 2019

Magma

[2] cat [(2^n +1 +(-1)^n)/2: n in [1..40]]; // G. C. Greubel, Oct 21 2019

Maple

spec:= [S,{S=Union(Sequence(Prod(Sequence(Z),Z)), Sequence(Prod(Z,Z)))}, unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);
seq(`if`(n=0, 2, (2^n +1 +(-1)^n)/2), n=0..40); # G. C. Greubel, Oct 21 2019

Mathematica

a[n_]:= (2^n +1 +(-1)^n +Boole[n==0])/2; (* Michael Somos, Jun 03 2014 *)
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 *)
LinearRecurrence[{2,1,-2}, {2,1,3,4}, 40] (* G. C. Greubel, Oct 21 2019 *)

PARI

{a(n)=(2^n+1+(-1)^n+(n==0))/2}; /* Michael Somos, Jun 03 2014 */

Sage

[2]+[(2^n +1 +(-1)^n)/2 for n in (1..40)] # G. C. Greubel, Oct 21 2019

Formula

G.f.: (2-3*x-x^2+x^3)/((1-x^2)*(1-2*x)).

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

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

From Paul Barry, Sep 18 2003: (Start)

a(n) = (2^n + 1 + (-1)^n + 0^n)/2.

E.g.f.: cosh(x)*(1+exp(x)). (End)

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

Extensions

More terms from James Sellers, Jun 05 2000