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

1,1

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.

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

Formula

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

Example

2.8900536382639638124570092961031296094359...

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 *)

Prog

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

Crossrefs

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

Sequence in context: A012649 A009627 A009677 * A021350 A016641 A155748

Adjacent sequences: A318602 A318603 A318604 * A318606 A318607 A318608

Keyword

nonn,cons

Author

A.H.M. Smeets, Sep 07 2018

Status

approved