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%I M2579 #82 Apr 13 2022 13:25:16
%S 1,3,6,14,33,71,150,318,665,1375,2830,5798,11825,24039,48742,98606,
%T 199113,401455,808382,1626038,3267809,6562295,13169814,26416318,
%U 52962681,106145855,212665582,425965126,853005201,1707833095,3418756806
%N Expansion of 1/((1-2x)(1+x^2)(1-x-2x^3)).
%C The number of simple squares in the biggest 'cloud' of the Harter-Heighway dragon of degree (n+4). Equals the number of double points in the biggest 'cloud' of the very same. - _Manfred Lindemann_, Dec 06 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 Unpublished, 1976.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="/A003477/b003477.txt">Table of n, a(n) for n = 0..1000</a>
%H D. E. Daykin, <a href="/A003229/a003229.pdf">Letter to N. J. A. Sloane, Mar 1974</a>
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H Kevin Ryde, <a href="http://user42.tuxfamily.org/dragon/index.html">Iterations of the Dragon Curve</a>, see index "BlobA".
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,5,-6,2,-4).
%F a(0) = 1; for n > 0, a(n) = 3*a(n-1) - 3*a(n-2) + 5*a(n-3) - 6*a(n-4) + 2*a(n-5) - 4*a(n-6) (where a(n)=0 for -5 <= n <= -1). - _Jon E. Schoenfield_, Apr 23 2010
%F From _Manfred Lindemann_, Dec 06 2015: (Start)
%F a(n) = 3*a(n-1) - 2*a(n-2) + 2*a(n-3) - 4*a(n-4) + Re(i^(n-4)), a(-5)=a(-4)=a(-3)=a(-2)=0 for all integers n element Z.
%F a(n+2)+a(n) = A003230(n+2)-A003230(n+1). [Daykin and Tucker equation (5)]
%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 BR:=1/(2*ROR-3), BC:=1/(2*ROC-3) and the zeros of (1-2*x) and (1+x^2) becomes
%F a(n) = (1/2)*(BR*ROR^-(n+4) + BC*ROC^-(n+4) + conjugate(BC*ROC^-(n+4)) + (2/5)*(1/2)^-(n+4) + (3/10 + i*(1/10))*i^-(n+4) + conjugate((3/10 + i*(1/10))*i^-(n+4))).
%F Simplified: a(n) = (BR/2)*ROR^-(n+4) + Re(BC*ROC^-(n+4)) + (1/5)*(1/2)^-(n+4) + Re((3/10 + i*(1/10))*i^-(n+4)).
%F (End)
%F Conjecture: a(n) = A077854(n) + 2*(a(n-3) + a(n-4) + ... + a(1)). - _Arie Bos_, Nov 29 2019
%p A003477:=1/(2*z-1)/(-1+z+2*z**3)/(1+z**2); # _Simon Plouffe_ in his 1992 dissertation
%p S:=series(1/((1-x-2*x^3)*(1-2*x)*(1+x^2)), x, 101): a:=n->coeff(S, x, n):
%p seq(a(n), n=0..100); # _Manfred Lindemann_, Dec 06 2015
%p a:= gfun:-rectoproc({a(n) = 3*a(n-1)-3*a(n-2)+5*a(n-3)-6*a(n-4)+2*a(n-5)-4*a(n-6),seq(a(i)=[1,3,6,14,33,71][i+1],i=0..5)},a(n),remember):
%p seq(a(n),n=0..100); # _Robert Israel_, Dec 14 2015
%t CoefficientList[Series[1/((1-2x)(1+x^2)(1-x-2x^3)),{x,0,40}],x] (* _Vincenzo Librandi_, Jun 11 2012 *)
%t LinearRecurrence[{3, -3, 5, -6, 2, -4}, {1, 3, 6, 14, 33, 71}, 31] (* _Arie Bos_, Dec 03 2019 *)
%o (PARI) Vec(1/((1-2*x)*(1+x^2)*(1-x-2*x^3))+O(x^99)) \\ _Charles R Greathouse IV_, Sep 26 2012
%Y Cf. A003230, A077949. - _Manfred Lindemann_, Dec 06 2015
%Y Cf. A077854.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
%E More terms from _Jon E. Schoenfield_, Apr 23 2010
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