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a(n) = 2^n - 2*(binomial(1,n) - binomial(0,n)).
3

%I #13 Jun 29 2023 13:08:14

%S 1,0,4,8,16,32,64,128,256,512,1024,2048,4096,8192,16384,32768,65536,

%T 131072,262144,524288,1048576,2097152,4194304,8388608,16777216,

%U 33554432,67108864,134217728,268435456,536870912,1073741824

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

%C Binomial transform is A186948.

%C Second binomial transform is A186947.

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

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

%H G. C. Greubel, <a href="/A186949/b186949.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (2).

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

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

%F E.g.f.: exp(2*x) - 2*x. - _G. C. Greubel_, Dec 01 2019

%p seq( `if`(n<2, 1-n, 2^n), n=0..30); # _G. C. Greubel_, Dec 01 2019

%t Table[If[n<2, 1-n, 2^n], {n, 0, 30}] (* _G. C. Greubel_, Dec 01 2019 *)

%o (PARI) vector(31, n, if(n<3, 2-n, 2^(n-1))) \\ _G. C. Greubel_, Dec 01 2019

%o (Magma) [n lt 2 select 1-n else 2^n: n in [0..30]]; // _G. C. Greubel_, Dec 01 2019

%o (Sage) [1,0]+[2^n for n in (2..30)] # _G. C. Greubel_, Dec 01 2019

%o (GAP) Concatenation([1,0], List([2..30], n-> 2^n )); # _G. C. Greubel_, Dec 01 2019

%K nonn,easy

%O 0,3

%A _Paul Barry_, Mar 01 2011