A309282 Decimal expansion of the circumference of a golden ellipse with a unit semi-major axis.

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

Offset: 1

Author

Amiram Eldar, Jul 05 2020

Keywords

• nonn
• cons

Comments

A golden ellipse is an ellipse inscribed in a golden rectangle. The concept of a golden ellipse was introduced by H. E. Huntley in 1970.

The aesthetic preferences of rectangles and ellipses with relation to the golden ratio were studied by Gustav Fechner in 1876. His results for ellipses were published by Witmer in 1893.

A golden ellipse with a semi-major axis 1 has a minor semi-axis 1/phi and an eccentricity 1/sqrt(phi), where phi is the golden ratio (A001622).

Examples

			5.154273178025879962492835539113341955287972235708661...
		

Links

• H. E. Huntley, The Divine Proportion: A Study in Mathematical Beauty, Dover, New York, 1970, page 65.
• H. E. Huntley, The Golden Ellipse, The Fibonacci Quarterly, Vol. 12, No. 1 (1974), pp. 38-40.
• Thomas Koshy, The Golden Ellipse and Hyperbola, in the book Fibonacci and Lucas Numbers with Applications, Wiley, 2001, chapter 26.
• M. C. Monzingo, A Note on the Golden Ellipse, The Fibonacci Quarterly, Vol. 14, No. 5 (1974), p. 388.
• A. D. Rawlins, A Note on the Golden Ratio, The Mathematical Gazette, Vol. 79, No. 484 (1995), p. 104.
• Stanislav Sýkora, Mathematical Constants, Stan's Library, Vol.II.
• Eric Weisstein's World of Mathematics, Ellipse.
• Wikipedia, Ellipse.
• Lightner Witmer, Zur experimentellen Aesthetik einfacher räumlicher Formverhältnisse, Philosophische Studien, Vol. 9 (1893), pp. 96-144.

Crossrefs

Cf. A001622 (phi), A094881 (area of the golden ellipse), A197762 (eccentricity of the golden ellipse).

Similar sequences: A138500, A274014.

Programs

Mathematica

RealDigits[4 * EllipticE[1/GoldenRatio], 10, 100][[1]]

Formula

Equals 4*E(1/phi), where E(x) is the complete elliptic integral of the second kind.