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%I #20 May 25 2024 15:34:17
%S 2,4,7,11,18,30,49,79,129,209,338,547,886,1434,2320,3754,6075,9830,
%T 15905,25735,41641,67376,109017,176394,285412,461806,747218,1209024
%N Curvature (rounded down) of the circle inscribed in the n-th golden triangle arranged in a spiral form.
%C 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.
%C The golden triangles created by this process are the same as the golden triangles inscribed in a logarithmic spiral.
%C 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.
%H Kival Ngaokrajang, <a href="/A228560/a228560.pdf">Illustration of initial terms</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Golden_triangle_(mathematics)">Golden triangle</a>.
%o (Small Basic)
%o phi=(1+Math.SquareRoot(5))/2
%o b[0]=phi
%o For n = 1 To 30
%o c=b[n-1]*(phi-1)
%o s=(2*b[n-1]+c)/2
%o r=math.SquareRoot((Math.Power((s-b[n-1]),2)*(s-c))/s)
%o b[n]=c
%o a=math.Floor(1/r)
%o TextWindow.Write(a+", ")
%o EndFor
%Y Cf. A001521 (for 45-45-90 triangles), A065565 (for 3:4:5 triangles), A014217.
%K nonn,more
%O 1,1
%A _Kival Ngaokrajang_, Aug 25 2013