A318605 Decimal expansion of geometric progression constant for Coxeter's Loxodromic Sequence of Tangent Circles.

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

Offset: 1

Author

A.H.M. Smeets, Sep 07 2018

Keywords

• nonn
• cons

Comments

This constant and its reciprocal are the real solutions of x^4 - 2*x^3 - 2*x^2 - 2*x + 1 = (x^2 - (sqrt(5)+1)*x + 1)*(x^2 + (sqrt(5)-1)*x + 1) = 0.

This constant and its reciprocal are the solutions of x^2 - (1+sqrt(5))*x + 1 = 0.

Decimal expansion of the largest x satisfying x^2 - (1+sqrt(5))*x + 1 = 0.

For sequences of type aa(n) = 2*(aa(n-1) + aa(n-2) + aa(n-3)) - aa(n-4) for arbitrary initial terms (except the trivial all zero), i.e., linear recurrence relations of order 4 with signature (2,2,2,-1), lim_{n -> infinity} aa(n)/aa(n-1) = this constant; see for instance A192234, A192237, A317973, A317974, A317975, A317976.

Examples

			2.8900536382639638124570092961031296094359...
		

Links

• A.H.M. Smeets, Table of n, a(n) for n = 1..9999 (terms 1..3000 from Muniru A Asiru)
• Index entries for algebraic numbers, degree 4

Crossrefs

Cf. A001622, A139339, A192234, A192237, A317973, A317974, A317975, A317976.

Programs

Maple

evalf[180]((1+sqrt(5))/2+sqrt((1+sqrt(5))/2)); # Muniru A Asiru, Nov 21 2018

Mathematica

RealDigits[GoldenRatio + Sqrt[GoldenRatio], 10 , 120][[1]] (* Amiram Eldar, Nov 22 2018 *)

PARI

((1+sqrt(5))/2 + sqrt((1+sqrt(5))/2)) \\ Michel Marcus, Nov 21 2018

Formula

Equals A001622 + A139339, i.e., phi + sqrt(phi) where phi is the golden ratio.