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A228560
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Curvature (rounded down) of the circle inscribed in the n-th golden triangle arranged in a spiral form.
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1
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2, 4, 7, 11, 18, 30, 49, 79, 129, 209, 338, 547, 886, 1434, 2320, 3754, 6075, 9830, 15905, 25735, 41641, 67376, 109017, 176394, 285412, 461806, 747218, 1209024
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graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,1
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COMMENTS
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Starting with a golden triangle whose base is of length 1 and whose sides are of length phi = (1+sqrt(5))/2, create the next golden triangle at the base of the previous triangle, i.e., sides length = 1 and base length = phi-1, and so on. a(n) is the floor of the curvature (inverse of the radius) of the circle inscribed in the n-th triangle.
The golden triangles created by this process are the same as the golden triangles inscribed in a logarithmic spiral.
The logarithmic spiral can be approximated by circular arcs of radii 1, phi-1, (phi-1)^2, ... which are the sides of bisected golden gnomons and center located at their related apex. The sequence whose n-th term is the curvature (rounded down) of the n-th such circular arc is A014217. See illustration in link.
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LINKS
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PROG
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(Small Basic)
phi=(1+Math.SquareRoot(5))/2
b[0]=phi
For n = 1 To 30
c=b[n-1]*(phi-1)
s=(2*b[n-1]+c)/2
r=math.SquareRoot((Math.Power((s-b[n-1]), 2)*(s-c))/s)
b[n]=c
a=math.Floor(1/r)
TextWindow.Write(a+", ")
EndFor
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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