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a(n) = Sum_{k=1..n} mu(k) * 2^(n - k).
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%I #22 May 19 2021 16:48:24

%S 0,1,1,1,2,3,7,13,26,52,105,209,418,835,1671,3343,6686,13371,26742,

%T 53483,106966,213933,427867,855733,1711466,3422932,6845865,13691730,

%U 27383460,54766919,109533837,219067673,438135346,876270693,1752541387,3505082775,7010165550

%N a(n) = Sum_{k=1..n} mu(k) * 2^(n - k).

%H Seiichi Manyama, <a href="/A344432/b344432.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: (Sum_{k>=1} mu(k) * x^k) / (1 - 2*x).

%F a(n) = 2 * a(n-1) + mu(n) for n > 0.

%F a(n) ~ A238270 * 2^n. - _Vaclav Kotesovec_, May 19 2021

%t a[n_] := Sum[MoebiusMu[k] * 2^(n-k), {k,1,n}]; Array[a, 40] (* _Amiram Eldar_, May 19 2021 *)

%o (PARI) a(n) = sum(k=1, n, moebius(k)*2^(n-k));

%o (PARI) my(N=40, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, moebius(k)*x^k)/(1-2*x)))

%o (PARI) a(n) = if(n==0, 0, 2*a(n-1)+moebius(n));

%Y Cf. A127513, A238270, A292524, A343425, A344433.

%K nonn

%O 0,5

%A _Seiichi Manyama_, May 19 2021