A186949 a(n) = 2^n - 2*(binomial(1,n) - binomial(0,n)).

1, 0, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824

Offset: 0

Author

Paul Barry, Mar 01 2011

Keywords

• nonn
• easy

Comments

Binomial transform is A186948.

Second binomial transform is A186947.

Inverse binomial transform is (-1)^n * A168277(n).

Essentially the same as A000079, A151821, A155559, A171449, and A171559.

Links

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

Programs

GAP

Concatenation([1,0], List([2..30], n-> 2^n )); # G. C. Greubel, Dec 01 2019

Magma

[n lt 2 select 1-n else 2^n: n in [0..30]]; // G. C. Greubel, Dec 01 2019

Maple

seq( `if`(n<2, 1-n, 2^n), n=0..30); # G. C. Greubel, Dec 01 2019

Mathematica

Table[If[n<2, 1-n, 2^n], {n, 0, 30}] (* G. C. Greubel, Dec 01 2019 *)

PARI

vector(31, n, if(n<3, 2-n, 2^(n-1))) \\ G. C. Greubel, Dec 01 2019

Sage

[1,0]+[2^n for n in (2..30)] # G. C. Greubel, Dec 01 2019

Formula

G.f.: (1 - 2*x + 4*x^2)/(1-2*x).

a(n) = Sum_{k=0..n} binomial(n,k)*(3^k - 2*k).

E.g.f.: exp(2*x) - 2*x. - G. C. Greubel, Dec 01 2019