A228560 Curvature of the circles (rounded down) inscribed in golden triangle arranged as 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 base length = 1 and sides length phi = (1+sqrt(5))/2, create the next golden triangle at the base of the previous step, i.e., sides length = 1 and base length = phi-1, and so on. a(n) is the floor of the curvature (inverse of radius) of the circle inscribed in each triangle.

The golden triangles created by this process are the same as the golden triangles inscribed in a logarithmic spiral.

The approximated logarithmic spiral can be formed by circle curvatures at the radii 1, phi-1, (phi-1)^2, ... which are the sides of bisected golden gnomons and center located at their related apex. The sequence of this circle curvature (rounded down) is A014217. See illustration in link.

Links

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

Wikipedia, Golden triangle

Kival Ngaokrajang, Illustration of initial terms

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

Author

Kival Ngaokrajang, Aug 25 2013

Status

approved