A318605 - OEIS

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

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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.

Ratio of radii of successive circles forming a recursive spiral of kissing circles about a point, with circle centers radius r * 1.08204... (A344362) from the spiral center point and consecutive circle centers 2.23703... (A195694) radians apart. - Charles L. Hohn, Nov 25 2025

LINKS

A.H.M. Smeets, Table of n, a(n) for n = 1..9999 (terms 1..3000 from Muniru A Asiru)

Charles L. Hohn, Recursive spiral of kissing circles about a point.

Index entries for algebraic numbers, degree 4

FORMULA

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

Largest real root of (Sum_{i=0..3} (1/x^i)) ^ 2 - 2 * Sum_{i=0..3} ((1/x^i)^2), from Descartes' theorem. See Comment. - Charles L. Hohn, Nov 25 2025

EXAMPLE

2.8900536382639638124570092961031296094359...

MAPLE

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

MATHEMATICA