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

0,1

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

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).

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

Example

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 + ...

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 *)

Prog

(PARI) {a(n)=(2^n+1+(-1)^n+(n==0))/2}; /* Michael Somos, Jun 03 2014 */
(Magma) [2] cat [(2^n +1 +(-1)^n)/2: n in [1..40]]; // G. C. Greubel, Oct 21 2019
(Sage) [2]+[(2^n +1 +(-1)^n)/2 for n in (1..40)] # G. C. Greubel, Oct 21 2019
(GAP) Concatenation([2], List([1..40], n-> (2^n +1 +(-1)^n)/2));  # G. C. Greubel, Oct 21 2019

Crossrefs

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

Sequence in context: A007444 A332309 A166476 * A086851 A001054 A218209

Adjacent sequences: A052947 A052948 A052949 * A052951 A052952 A052953

Keyword

easy,nonn

Author

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

Extensions

More terms from James A. Sellers, Jun 05 2000

Status

approved