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A094024 Alternating 1 with one less than the powers of 2.

%I A094024 #28 Sep 08 2022 08:45:13
%S A094024 1,1,1,3,1,7,1,15,1,31,1,63,1,127,1,255,1,511,1,1023,1,2047,1,4095,1,
%T A094024 8191,1,16383,1,32767,1,65535,1,131071,1,262143,1,524287,1,1048575,1,
%U A094024 2097151,1,4194303,1,8388607,1,16777215,1,33554431,1,67108863,1
%N A094024 Alternating 1 with one less than the powers of 2.
%C A094024 Inverse binomial transform of A052542. Partial sums are A075427.
%C A094024 Let F(x) = product {n >= 0} (1 - x^(3*n+1))/(1 - x^(3*n+2)). This sequence is the simple continued fraction expansion of the real number F(1/2) = 0.64227 25013 85234 96714 ... = 1/(1 + 1/(1 + 1/(1 + 1/(3 + 1/(1 + 1/(7 + 1/(1 + 1/(15 + ...)))))))). See A111317. - _Peter Bala_, Dec 26 2012
%H A094024 Vincenzo Librandi, <a href="/A094024/b094024.txt">Table of n, a(n) for n = 0..5000</a>
%H A094024 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (-1,2,2).
%F A094024 G.f.: (1+2*x) / ((1+x) * (1-2*x^2)).
%F A094024 E.g.f.: exp(-x) + 2*sinh(sqrt(2)*x) / sqrt(2).
%F A094024 a(n) = (-1)^n + ((sqrt(2))^n - (-sqrt(2))^n) / sqrt(2).
%F A094024 a(n) = (1-(-1)^n) * 2^floor(n/2) + (-1)^n. - _Ralf Stephan_, Aug 19 2013
%F A094024 a(n) = -a(n-1) + 2*a(n-2) + 2*a(n-3). - _Andrew Howroyd_, Feb 21 2018
%t A094024 LinearRecurrence[{-1, 2, 2}, {1, 1, 1}, 60] (* _Jean-François Alcover_, Jul 02 2018 *)
%o A094024 (Magma) [Ceiling((-1)^n+((Sqrt(2))^n-(-Sqrt(2))^n)/Sqrt(2)): n in [0..50]]; // _Vincenzo Librandi_, Aug 17 2011
%o A094024 (PARI) a(n)=(1-(-1)^n)*2^floor(n/2)+(-1)^n
%Y A094024 Cf. A000129, A016116.
%Y A094024 Cf. A111317.
%K A094024 easy,nonn
%O A094024 0,4
%A A094024 _Paul Barry_, Apr 22 2004
%E A094024 Better name from _Ralf Stephan_, Aug 19 2013
%E A094024 Even terms for n >= 60 corrected in b-file by _Andrew Howroyd_, Feb 21 2018