A362566 - OEIS

%I #29 May 04 2023 06:40:14

%S 0,1,2,6,15,42,77,180,345,806,1457,3276,5985,13462,24257,54060,97665,

%T 217686,391937,871596,1570305,3492182,6286337,13972140,25155585,

%U 55911766,100642817,223660716,402612225,894735702,1610530817,3578997420,6442287105,14316361046

%N a(n) is the area of the smallest rectangle that the Harter-Heighway Dragon Curve will fit in after n doublings, starting with a segment of length 1.

%C When constructing this sequence, the rectangles that are considered are those whose sides are parallel to the corresponding links of the dragon curve.

%H Nicolay Avilov, <a href="/A362566/a362566.jpg">Illustration of initial terms</a>.

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,-3,8,-8,-12,12,-16,16)

%F From _Andrey Zabolotskiy_, _Joerg Arndt_ and _Kevin Ryde_, May 03 2023: (Start)

%F G.f.: x * (1 + x + x^2 + 6*x^3 + 7*x^4 + 2*x^6) / ((1 - x) * (1 - 2*x) * (1 + 2*x) * (1 + x^2) * (1 - 2*x^2) * (1 + 2*x^2)).

%F a(n) =

%F   (3*2^n -  5*2^(n/2)     + 2) / 2 for n == 0 (mod 2),

%F   (5*2^n -  9*2^((n-1)/2) + 2) / 3 for n == 1 (mod 4),

%F   (5*2^n - 13*2^((n-1)/2) + 4) / 3 for n == 3 (mod 4). (End)

%e See link:

%e a(3) = 2*3 = 6;

%e a(4) = 3*5 = 15;

%e a(5) = 6*7 = 42.

%o (Python)

%o x1, x2, y1, y2, ex, ey, a = 0, 1, 0, 0, 1, 0, [0]

%o for n in range(40):

%o     ex, ey = ex-ey, ey+ex

%o     x1r, x2r, y1r, y2r = y1+ex, y2+ex, -x2+ey, -x1+ey

%o     x1, x2, y1, y2 = min(x1, x1r), max(x2, x2r), min(y1, y1r), max(y2, y2r)

%o     a.append((x2-x1)*(y2-y1))

%o print(a) # _Andrey Zabolotskiy_, May 03 2023

%Y Cf. A014577, A341029.

%K nonn,easy

%O 0,3

%A _Nicolay Avilov_, Apr 25 2023

%E Terms a(16) and beyond and a(0)=0 from _Andrey Zabolotskiy_, Apr 27 2023