A385446 Decimal expansion of -7 + 10*phi, with the golden section phi = A001622.

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

Offset: 1

Author

Wolfdieter Lang, Jul 01 2025

Keywords

• nonn
• cons
• easy

Comments

This constant d gives the imaginary part of -2*11*Z = c + d*i, where Z is the fixed point of a complex function w (of the loxodromic type) mapping vertices of golden triangles, starting with vertices (D_1, D_2, D_3), circumcribed by the unit circle with center at the origin, and D_1 = i (the complex unit), 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*i/5) = -((2 - phi) + s*i)/2 and b = (1 - phi)*i, where s = sqrt(3 - phi) = A182007 (the length of the base (D2, D3) of the first triangle).

The real part c = (-1 + 3*phi)*s is given in A385445.

For details see A385445, and eqs.(5a,b) of the linked paper there.

Examples

			9.18033988749894848204586834365638117720309179805762862...
		

Crossrefs

Cf. A001622, A182007, A385445.

Programs

Mathematica

RealDigits[10*GoldenRatio - 7, 10, 120][[1]] (* Amiram Eldar, Jul 02 2025 *)

Formula

Equals -7 + 10*phi, an integer in the quadratic number field Q(sqrt(5)).

Equals 10*A176055-12 = 10*A104457-17 = 10*A001622-7 . - R. J. Mathar, Jul 06 2025