A242671 Decimal expansion of k2, a Diophantine approximation constant such that the area of the "critical parallelogram" (in this case a square) is 4*k2.

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

Offset: 0

Author

Jean-François Alcover, May 20 2014

Keywords

• nonn
• cons
• easy

Comments

Quoting Steven Finch: "The slopes of the 'critical parallelogram' are (1+sqrt(5))/2 [phi] and (1-sqrt(5))/2 [-1/phi]."

Essentially the same as A229780, A134972, A134945, A098317 and A002163. - R. J. Mathar, May 23 2014

Let W_n be the collection of all binary words of length n that do not contain two consecutive 0's. Let r_n be the ratio of the total number of 1's in W_n divided by the total number of letters in W_n. Then lim_{n->oo} r_n = 0.723606... Equivalently, lim_{n->oo} A004798(n)/(n*A000045(n+2)) = 0.723606... - Geoffrey Critzer, Feb 04 2022

The limiting frequency of the digit 0 in the base phi representation of real numbers in the range [0,1], where phi is the golden ratio (A001622) (Rényi, 1957). - Amiram Eldar, Mar 18 2025

Examples

			k2 = 0.723606797749978969640917366873127623544...
		

References

• Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.23, p. 176.

Links

• Alfréd Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar., Vol.8, No. 3-4 (1957), pp. 477-493.
• Index entries for algebraic numbers, degree 2.

Crossrefs

Cf. A000032, A000045, A001622, A002163, A004798, A020762, A081007, A094214, A094874, A098317, A134972, A229780, A244847, A296184, A300074, A344212.

Programs

Mathematica

RealDigits[(1+1/Sqrt[5])/2, 10, 100] // First

PARI

(1 + 1/sqrt(5))/2 \\ Stefano Spezia, Dec 07 2024

Formula

Equals (1 + 1/sqrt(5))/2.

Equals 1/A094874. - Michel Marcus, Dec 01 2018

From Amiram Eldar, Feb 11 2022: (Start)

Equals phi/sqrt(5), where phi is the golden ratio (A001622).

Equals lim_{k->oo} Fibonacci(k+1)/Lucas(k). (End)

From Amiram Eldar, Nov 28 2024: (Start)

Equals A344212/2 = A296184/5 = A300074^2 = sqrt(A229780).

Equals Product_{k>=1} (1 - 1/A081007(k)). (End)

Equals 1 - A244847. - Amiram Eldar, Mar 18 2025