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%I #39 Sep 08 2022 08:45:03
%S 0,3,3,9,15,33,63,129,255,513,1023,2049,4095,8193,16383,32769,65535,
%T 131073,262143,524289,1048575,2097153,4194303,8388609,16777215,
%U 33554433,67108863,134217729,268435455,536870913,1073741823,2147483649
%N a(n) = 2^n + (-1)^(n+1).
%C The identity 2 = 2^2/3 + 2^3/(3*3) - 2^4/(3*3*9) - 2^5/(3*3*9*15) + + - - can be viewed as a generalized Engel-type expansion of the number 2 to the base 2. Compare with A014551. - _Peter Bala_, Nov 13 2013
%D D. M. Burton, Elementary Number Theory, Allyn and Bacon, Inc. Boston, MA, 1976, p. 29.
%H Vincenzo Librandi, <a href="/A062510/b062510.txt">Table of n, a(n) for n = 0..1000</a>
%H G. Everest, Y. Puri and T. Ward, <a href="https://arxiv.org/abs/math/0204173">Integer sequences counting periodic points</a>, arXiv:math/0204173 [math.NT], 2002.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,2).
%F a(n) = 3*A001045(n). - _Paul Curtz_, Jan 17 2008
%F G.f.: 3*x / ( (1+x)*(1-2*x) )
%F G.f.: Q(0) where Q(k)= 1 - 1/(4^k - 2*x*16^k/(2*x*4^k - 1/(1 + 1/(2*4^k - 8*x*16^k/(4*x*4^k + 1/Q(k+1) ))))); (continued fraction). - _Sergei N. Gladkovskii_, Apr 13 2013
%F E.g.f.: (exp(3*x) - 1)*exp(-x). - _Ilya Gutkovskiy_, Nov 20 2016
%t LinearRecurrence[{1,2},{0,3}, 30] (* or *) Table[2^n - (-1)^n, {n,0,30}] (* _G. C. Greubel_, Jan 15 2018 *)
%o (PARI) for(n=0,22,print(2^n+(-1)^(n+1)))
%o (Magma) [2^n + (-1)^(n+1): n in [0..40]]; // _Vincenzo Librandi_, Aug 14 2011
%Y Cf. A102345, A105723.
%K easy,nonn
%O 0,2
%A _Jason Earls_, Jun 24 2001
%E More terms from Larry Reeves (larryr(AT)acm.org), Jul 06 2001
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