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

%I A081654 #41 Aug 27 2025 08:02:27
%S A081654 1,8,32,128,512,2048,8192,32768,131072,524288,2097152,8388608,
%T A081654 33554432,134217728,536870912,2147483648,8589934592,34359738368,
%U A081654 137438953472,549755813888,2199023255552,8796093022208,35184372088832
%N A081654 a(n) = 2*4^n - 0^n.
%C A081654 Binomial transform of A081632. Inverse binomial transform of A081655.
%H A081654 Vincenzo Librandi, <a href="/A081654/b081654.txt">Table of n, a(n) for n = 0..1000</a>
%H A081654 <a href="/index/Di#divseq">Index to divisibility sequences</a>
%H A081654 <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (4).
%F A081654 a(0)=1, a(n) = 2*4^n, n>0
%F A081654 G.f.: (1+4*x)/(1-4*x).
%F A081654 E.g.f. 2*exp(4*x)-1.
%F A081654 With interpolated zeros, this is 2^n - 0^n + (-2)^n. - _Paul Barry_, Sep 06 2003
%F A081654 a(n) = A081294(n+1), n>0. - _R. J. Mathar_, Sep 17 2008
%F A081654 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
%e A081654 a(0) = 2*4^0 - 0^0 = 2 - 1 = 1 (use 0^0 = 1).
%t A081654 CoefficientList[Series[(1 + 4 x) / (1 - 4 x), {x, 0, 40}], x] (* _Vincenzo Librandi_, Aug 10 2013 *)
%o A081654 (PARI) a(n)=2*4^n-0^n \\ _Charles R Greathouse IV_, Apr 09 2012
%o A081654 (Magma) [2*4^n-0^n: n in [0..30]]; // _Vincenzo Librandi_, Aug 10 2013
%o A081654 (PARI) x='x+O('x^100); Vec((1+4*x)/(1-4*x)) \\ _Altug Alkan_, Dec 14 2015
%Y A081654 Cf. A000244 (3^n), A187093.
%Y A081654 Essentially the same as A004171.
%K A081654 easy,nonn,changed
%O A081654 0,2
%A A081654 _Paul Barry_, Mar 26 2003