A362566 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.

0, 1, 2, 6, 15, 42, 77, 180, 345, 806, 1457, 3276, 5985, 13462, 24257, 54060, 97665, 217686, 391937, 871596, 1570305, 3492182, 6286337, 13972140, 25155585, 55911766, 100642817, 223660716, 402612225, 894735702, 1610530817, 3578997420, 6442287105, 14316361046

Offset: 0

Author

Nicolay Avilov, Apr 25 2023

Keywords

• nonn
• easy

Comments

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

Examples

			See link:
a(3) = 2*3 = 6;
a(4) = 3*5 = 15;
a(5) = 6*7 = 42.
		

Links

• Nicolay Avilov, Illustration of initial terms.
• Index entries for linear recurrences with constant coefficients, signature (1,3,-3,8,-8,-12,12,-16,16)

Crossrefs

Cf. A014577, A341029.

Programs

Python

x1, x2, y1, y2, ex, ey, a = 0, 1, 0, 0, 1, 0, [0]
for n in range(40):
    ex, ey = ex-ey, ey+ex
    x1r, x2r, y1r, y2r = y1+ex, y2+ex, -x2+ey, -x1+ey
    x1, x2, y1, y2 = min(x1, x1r), max(x2, x2r), min(y1, y1r), max(y2, y2r)
    a.append((x2-x1)*(y2-y1))
print(a) # Andrey Zabolotskiy, May 03 2023

Formula

From Andrey Zabolotskiy, Joerg Arndt and Kevin Ryde, May 03 2023: (Start)

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)).

a(n) =

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

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

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

Extensions

Terms a(16) and beyond and a(0)=0 from Andrey Zabolotskiy, Apr 27 2023