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A227036 Expansion of 2*(1+x^2)/((1-x)*(1-x-2*x^3)).

%I A227036 #35 Jul 29 2024 06:17:35
%S A227036 2,4,8,16,28,48,84,144,244,416,708,1200,2036,3456,5860,9936,16852,
%T A227036 28576,48452,82160,139316,236224,400548,679184,1151636,1952736,
%U A227036 3311108,5614384,9519860,16142080,27370852,46410576,78694740,133436448,226257604,383647088,650519988,1103035200,1870329380
%N A227036 Expansion of 2*(1+x^2)/((1-x)*(1-x-2*x^3)).
%C A227036 Conjecture: The perimeter of the n-th iteration of the Harter-Heighway dragon is a(n) segments or a(n)/2^(n/2) base units.
%C A227036 a(n) = 2^(n+1)-4*A003230(n-4) (two times the number of segments, minus four times the number of squares)
%C A227036 The first differences 2, 2, 4, 8, 12, 20,.. are twice the (empirical) A203175. - _R. J. Mathar_, Jul 02 2013
%H A227036 Vincenzo Librandi, <a href="/A227036/b227036.txt">Table of n, a(n) for n = 0..1000</a>
%H A227036 Kevin Ryde, <a href="http://user42.tuxfamily.org/dragon/index.html">Iterations of the Dragon Curve</a>, see index "B" and "R".
%H A227036 Helena Verrill, <a href="https://arxiv.org/abs/2407.17326">On the Boundary of the Harter-Heighway dragon curve</a>, arXiv:2407.17326 [math.CO], 2024.
%H A227036 Wikipedia, <a href="http://en.wikipedia.org/wiki/Dragon_curve">Dragon curve</a>
%H A227036 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,2,-2).
%e A227036 For the 4th iteration, take two 3rd iteration dragons (2*16); put together, they will make one square, so subtract the inner perimeter 4.
%t A227036 LinearRecurrence[{2, -1, 2, -2}, {2, 4, 8, 16}, 40] (* _T. D. Noe_, Jul 02 2013 *)
%t A227036 CoefficientList[Series[2 (1 + x^2) / ((1 - x) (1 - x - 2 x^3)), {x, 0, 40}], x] (* _Vincenzo Librandi_, Jul 17 2013 *)
%o A227036 (PARI) Vec(2*(1+x^2)/((1-x)*(1-x-2*x^3))+O(x^66)) \\ _Joerg Arndt_, Jul 01 2013
%Y A227036 Cf. A014577.
%K A227036 nonn,easy
%O A227036 0,1
%A A227036 _Roland Kneer_, Jun 28 2013