A081654 a(n) = 2*4^n - 0^n.

1, 8, 32, 128, 512, 2048, 8192, 32768, 131072, 524288, 2097152, 8388608, 33554432, 134217728, 536870912, 2147483648, 8589934592, 34359738368, 137438953472, 549755813888, 2199023255552, 8796093022208, 35184372088832

Offset: 0

Author

Paul Barry, Mar 26 2003

Keywords

• easy
• nonn
• changed

Comments

Binomial transform of A081632. Inverse binomial transform of A081655.

Examples

			a(0) = 2*4^0 - 0^0 = 2 - 1 = 1 (use 0^0 = 1).
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index to divisibility sequences
• Index entries for linear recurrences with constant coefficients, signature (4).

Crossrefs

Cf. A000244 (3^n), A187093.

Essentially the same as A004171.

Programs

Magma

[2*4^n-0^n: n in [0..30]]; // Vincenzo Librandi, Aug 10 2013

Mathematica

CoefficientList[Series[(1 + 4 x) / (1 - 4 x), {x, 0, 40}], x] (* Vincenzo Librandi, Aug 10 2013 *)

PARI

a(n)=2*4^n-0^n \\ Charles R Greathouse IV, Apr 09 2012

PARI

x='x+O('x^100); Vec((1+4*x)/(1-4*x)) \\ Altug Alkan, Dec 14 2015

Formula

a(0)=1, a(n) = 2*4^n, n>0

G.f.: (1+4*x)/(1-4*x).

E.g.f. 2*exp(4*x)-1.

With interpolated zeros, this is 2^n - 0^n + (-2)^n. - Paul Barry, Sep 06 2003

a(n) = A081294(n+1), n>0. - R. J. Mathar, Sep 17 2008

For n>0, a(n) = 2 * (1 + 3^(n-1) + Sum{x=1..n-2}Sum{k=0..x-1}(binomial(x-1,k)*(3^(k+1) + 3^(n-x+k)))). - J. Conrad, Dec 10 2015