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%I #53 Jun 29 2023 11:48:19
%S 2,3,6,12,24,48,96,192,384,768,1536,3072,6144,12288,24576,49152,98304,
%T 196608,393216,786432,1572864,3145728,6291456,12582912,25165824,
%U 50331648,100663296,201326592,402653184,805306368,1610612736,3221225472,6442450944
%N Row sums of the Lucas triangle A029635.
%C Map a binary sequence b=[ b_1,...] to a binary sequence c=[ c_1,...] so that C = 1/Product((1-x^i)^c_i == 1+Sum b_i*x^i mod 2.
%C This produces 2 new sequences: d={i:c_i=1} and e=[ 1,e_1,... ] where C = 1 + Sum e_i*x^i.
%C This sequence is d when b=[ 0,1,1,1,1,...].
%C Number of rises after n+1 iterations of morphism A007413.
%C a(n) written in base 2: a(0) = 10, a(n) for n >= 1: 11, 110, 11000, 110000, ..., i.e.: 2 times 1, (n-1) times 0 (see A003953(n)). - _Jaroslav Krizek_, Aug 17 2009
%C Row sums of the Lucas triangle A029635. - _Sergio Falcon_, Mar 17 2014
%H Vincenzo Librandi, <a href="/A042950/b042950.txt">Table of n, a(n) for n = 0..1000</a>
%H S. Kitaev and T. Mansour, <a href="http://arxiv.org/abs/math/0210170">Counting the occurrences of generalized patterns in words generated by a morphism</a>, arXiv:math/0210170 [math.CO], 2002.
%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (2).
%F G.f.: (2-x)/(1-2*x).
%F a(n) = 2*a(n-1), n > 1; a(0)=2, a(1)=3.
%F a(n) = A003945(n), for n > 0.
%F From _Paul Barry_, Dec 06 2004: (Start)
%F Binomial transform of 2, 1, 2, 1, 2, 1, ... = (3+(-1)^n)/2.
%F a(n) = (3*2^n + 0^n)/2. (End)
%F a(0) = 2, a(n) = 3*2^(n-1) = 2^n + 2^(n-1) for n >= 1. - _Jaroslav Krizek_, Aug 17 2009
%F a(n) = 2^(n+1) - 2^(n-1), for n > 0. - _Ilya Gutkovskiy_, Aug 08 2015
%t Table[ Ceiling[3*2^(n - 1)], {n, 0, 32}] (* _Robert G. Wilson v_, Jul 08 2006 *)
%t a[0] = 2; a[1] = 3; a[n_] := 2a[n - 1]; Table[a[n], {n, 0, 32}] (* _Robert G. Wilson v_, Jul 08 2006 *)
%t f[s_] := Append[s, 1 + Plus @@ s]; Nest[f, {2}, 32] (* _Robert G. Wilson v_, Jul 08 2006 *)
%t CoefficientList[Series[(2 - x)/(1 - 2x), {x, 0, 32}], x] (* _Robert G. Wilson v_, Jul 08 2006 *)
%o (PARI) a(n)=ceil(3*2^(n-1))
%o (Magma) [2] cat [2^(n+1) - 2^(n-1): n in [1..40]]; // _Vincenzo Librandi_, Aug 08 2015
%Y Cf. A007283.
%K nonn,easy
%O 0,1
%A _N. J. A. Sloane_ and _J. H. Conway_
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