A228560 Curvature (rounded down) of the circle inscribed in the n-th golden triangle arranged in a spiral form.

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

Offset

1,1

Comments

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.

Links

Table of n, a(n) for n=1..28.

Kival Ngaokrajang, Illustration of initial terms.

Wikipedia, Golden triangle.

Prog

(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

Crossrefs

Cf. A001521 (for 45-45-90 triangles), A065565 (for 3:4:5 triangles), A014217.

Sequence in context: A004696 A293418 A289077 * A018063 A289004 A000570

Adjacent sequences: A228557 A228558 A228559 * A228561 A228562 A228563

Keyword

nonn,more

Author

Kival Ngaokrajang, Aug 25 2013

Status

approved