A084215 Expansion of g.f.: (1+x^2)/(1-2*x).

1, 2, 5, 10, 20, 40, 80, 160, 320, 640, 1280, 2560, 5120, 10240, 20480, 40960, 81920, 163840, 327680, 655360, 1310720, 2621440, 5242880, 10485760, 20971520, 41943040, 83886080, 167772160, 335544320, 671088640, 1342177280, 2684354560, 5368709120, 10737418240

Offset: 0

Author

Paul Barry, May 19 2003

Keywords

• easy
• nonn

Comments

Associated with a math magic problem.

Elements are the sums of consecutive pairs of elements of A084214.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2).

Crossrefs

Cf. A020714, A060816, A084214.

Programs

Magma

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+x^2)/(1-2*x))); // G. C. Greubel, Oct 08 2018

Mathematica

Join[{1, 2, a = 5}, Table[a = 2*a, {n,0,40}]] (* Vladimir Joseph Stephan Orlovsky, Jun 09 2011 *)
Table[Int[2^(n-2)*5],{n,0,40}] (* Taher Jamshidi, Sep 15 2012 *)
CoefficientList[Series[(1 + x^2)/(1 - 2 x), {x, 0, 30}], x] (* G. C. Greubel, Oct 08 2018 *)

PARI

x='x+O('x^30); Vec((1+x^2)/(1-2*x)) \\ G. C. Greubel, Oct 08 2018

Formula

a(n) = Sum_{k=0..n} 2^(n-k)*binomial(1, k/2)*(1+(-1)^k)/2. - Paul Barry, Oct 15 2004

a(n) = A020714(n-2), n > 1. - R. J. Mathar, Dec 19 2008

From Gary W. Adamson, Aug 26 2011: (Start)

a(n) is the sum of top row terms of M^n, M is an infinite square production matrix as follows:

1, 1, 0, 0, 0, 0, ...

1, 1, 1, 0, 0, 0, ...

0, 0, 0, 0, 0, 0, ...

0, 0, 0, 0, 0, 0, ...

...

E.g.: a(4) = 20 = (8 + 8 + 4) since the top row of M^4 = (8, 8, 4, 0, 0, 0, ...). (End)

a(n) = floor(2^(n-2)*5). - Taher Jamshidi, Sep 15 2012

a(n) = 2*a(n-1) for n >= 3, a(0) = 1, a(1) = 2, a(2) = 5. - Philippe Deléham, Mar 13 2013

E.g.f.: (5*exp(2*x) - 2*x - 1)/4. - Stefano Spezia, Feb 20 2023