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A003230 Expansion of 1/((1-x)*(1-2*x)*(1-x-2*x^3)).
(Formerly M3417)
4

%I M3417

%S 1,4,11,28,67,152,335,724,1539,3232,6727,13900,28555,58392,118959,

%T 241604,489459,989520,1997015,4024508,8100699,16289032,32726655,

%U 65705268,131837763,264399936,530028199,1062139180,2127809963

%N Expansion of 1/((1-x)*(1-2*x)*(1-x-2*x^3)).

%C The number of simple squares in the (n+4)-th iteration of the Harter-Heighway dragon (see Wikipedia reference below). - _Roland Kneer_, Jul 01 2013

%C The number of double points of the (n+4)-th iteration of the Harter-Heighway dragon. - _Manfred Lindemann_, Nov 11 2015

%D D. E. Daykin and S. J. Tucker, Introduction to Dragon Curves. Unpublished, 1976. See links in A003229 for an earlier version.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Vincenzo Librandi, <a href="/A003230/b003230.txt">Table of n, a(n) for n = 0..1000</a>

%H D. E. Daykin, <a href="/A003229/a003229_1.pdf">Letter to N. J. A. Sloane, Dec 1973</a>

%H D. E. Daykin, <a href="/A003229/a003229.pdf">Letter to N. J. A. Sloane, Mar 1974</a>

%H Simon Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992.

%H Simon Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">1031 Generating Functions and Conjectures</a>, Université du Québec à Montréal, 1992.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Dragon_curve">Dragon curve: Harter-Heighway dragon</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,4,-6,4).

%F a(n+3) = a(n+2) + 2*a(n) + 2^(n+4) - 1, with a(-3)=a(-2)=a(-1)=0. - _Manfred Lindemann_, Nov 11 2015

%F a(n+2) - a(n+1) = A003477(n+2) + A003477(n). - _Manfred Lindemann_, Dec 08 2015

%F a(n) = q(n) + q(n-1) + 2*Sum_{i=0..n-2}(q(i)), where q(i)=A003477 and q(-1)=0. - _Manfred Lindemann_, Dec 08 2015

%F From _Manfred Lindemann_, Nov 11 2015: (Start)

%F With thrt:=(54+6*sqrt(87))^(1/3), ROR:=(thrt/6-1/thrt) and RORext:=(thrt/6+1/thrt) becomes ROC:=(1/2)*(i*sqrt(3)*RORext-ROR), where i^2=-1.

%F Now ROR, ROC and conjugate(ROC) are the zeros of 1-x-2*x^3.

%F With AR:=(2*ROR^2+ROR+2)/(2*ROR-3), AC:=(2*ROC^2+ROC+2)/(2*ROC-3) and the zeros of (1-2*x) and (1-x)

%F a(n) = (1/2)*(AR*ROR^-(n+4)+AC*ROC^-(n+4)+conjugate(AC*ROC^-(n+4))+1*(1/2)^-(n+4)+1*1^-(n+4)).

%F Simplified: a(n) = (1/2)*(AR*ROR^-(n+4)+2*Re(AC*ROC^-(n+4))+2^(n+4)+1).

%F (End)

%p A003230:=-1/(z-1)/(2*z-1)/(-1+z+2*z**3); # _Simon Plouffe_ in his 1992 dissertation

%p S:=series(1/((1-x)*(1-2*x)*(1-x-2*x^3)),x,101): a:=n->coeff(S,x,n):

%p seq(a(n),n=0..100); # _Manfred Lindemann_, Nov 13 2015

%t CoefficientList[Series[1/((1-x)*(1-2x)*(1-x-2x^3)),{x,0,40}],x] (* _Vincenzo Librandi_, Jun 11 2012 *)

%o (PARI) Vec(1/((1-x)*(1-2*x)*(1-x-2*x^3))+O(x^66)) \\ _Joerg Arndt_, Jun 29 2013

%Y Cf. A003229, A077949, A003477.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_

%E More terms from _James A. Sellers_, Aug 21 2000

%E Maple program corrected by _Robert Israel_, Nov 11 2015

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Last modified October 16 15:48 EDT 2019. Contains 328101 sequences. (Running on oeis4.)