A385445 - OEIS

A385445

Decimal expansion of (-1 + 3*phi)*sqrt(3 - phi), with the golden section phi = A001622.

4, 5, 3, 0, 7, 6, 8, 5, 9, 3, 1, 8, 5, 9, 7, 5, 1, 7, 4, 3, 6, 1, 2, 2, 4, 0, 9, 0, 9, 9, 8, 1, 4, 7, 3, 2, 3, 2, 3, 8, 8, 8, 6, 9, 2, 9, 4, 6, 8, 2, 0, 9, 3, 5, 2, 5, 3, 9, 2, 8, 8, 9, 0, 5, 0, 6, 6, 3, 6, 2, 0, 7, 2, 1, 8, 6, 4, 5, 7, 0, 9, 5, 2, 9

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OFFSET

1,1

COMMENTS

This constant c gives the real part of -2*11*Z = (c + d*i), where Z is the (finite) fixed point of a complex function w (of the loxodromic type) mapping iteratively the vertices of golden triangles, starting with vertices (D_1, D_2, D_3), circumscribed by the unit circle with center at the origin, and D_1 = i, D_2 = (s - phi*i)/2 and D_3 = (-s - phi*i)/2. This function is w(z) = a*z + b, with a = (-1 + phi) * exp(-(3*Pi/5)*i) = -((2 - phi) + s*i)/2 and b = (1 - phi)*i, where s = sqrt(3 - phi) = A182007 (the length of the base (D_2, D_3) of the first triangle, also called s_1).

The imaginary part of -2*11*Z is d = -7 + 10*phi = A385446.

If the fixed point Z = -(0.20594... + 0.41728...*i) is chosen as origin then the loxodromic map is W(z') = a*z' (where z' = z - Z and W(z') = w(z'+Z) - Z).

For details see the linked paper, eqs.(5a,b) for w(z), eq.(6) for Z and eq.(7) for W(z'). (In eq.(5b) the i is missing in the exponent.) The nesting of golden triangles as shown in Fig. 1 of the link leads to the fixed point Z.

The vertices of the nested golden triangles can be connected by a spiral built of circular sections with angle 108 degrees, centered at vertices D_{n+3} and shrinking radii s_n =(- 1 + phi)^(n-1)*s. Note that the curvature of this spiral is not continuous.

The length(Z, D_n) =: rho_n of the spokes of the spiral is (-1 + phi)^(n-1)*rho_1, with sqrt(11)*rho_1 = sqrt(8 + 9*phi) = sqrt(5 + 7*phi)*s = A385447.

For the length ratio rho_1/s see A385448.

The logarithmic spiral connecting the vertices D_{n+1} is given in polar coordinates by rho(Phi) = rho_1 * exp((-(5/(3*Pi)) * log(phi)*Phi), with the vertices obtained in polar coordinates for Phi = Phi_n = (3*Pi/5)*n, namely rho(Phi_n) = rho_{n+1}, for n >= 0. For log(phi) see A002390. Note that the nonnegative x-axis is now along Z, D_1. The angle(Z, D1, D4) =: gamma is given by arctan((18 - 11*phi)/s) = arcsin(rho_4 / 2) = 0.169860704... or 9.7323... degrees. See Fig. 3 of the linked paper.

LINKS

Table of n, a(n) for n=1..84.

Wolfdieter Lang, On a Conformal Mapping of Golden Triangles, Papua New Guinea Journal of Maths. 1991, Vol.2, No:2, pp. 12 - 18 (with corrections).

Index entries for algebraic numbers, degree 4.

FORMULA

Equals (-1 + 3*phi)*sqrt(3 - phi) = (A090550 - 2)*A182007.

Equals sqrt(27 - 4*phi).

EXAMPLE

4.5307685931859751743612240909981473232388869294682093...

MATHEMATICA

RealDigits[Sqrt[27 - 4*GoldenRatio], 10, 120][[1]] (* Amiram Eldar, Jul 02 2025 *)

PROG