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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.
2
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
0,3
COMMENTS
When constructing this sequence, the rectangles that are considered are those whose sides are parallel to the corresponding links of the dragon curve.
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)
EXAMPLE
See link:
a(3) = 2*3 = 6;
a(4) = 3*5 = 15;
a(5) = 6*7 = 42.
PROG
(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
CROSSREFS
Sequence in context: A348012 A139379 A236110 * A280782 A307308 A065178
KEYWORD
nonn,easy
AUTHOR
Nicolay Avilov, Apr 25 2023
EXTENSIONS
Terms a(16) and beyond and a(0)=0 from Andrey Zabolotskiy, Apr 27 2023
STATUS
approved