A289917 Decimal expansion of the limiting ratio of consecutive terms of A289916.

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

Offset

1,2

Comments

Ratio of radii of successive spheres forming a recursive spiral of kissing spheres tangent to a plane, with sphere centers radius r * 1.05328077... from the spiral axis and consecutive sphere centers 2.28020842... radians apart. - Charles L. Hohn, Nov 13 2025

Links

Table of n, a(n) for n=1..86.

Charles L. Hohn, Recursive spiral of kissing spheres tangent to a plane.

Index entries for algebraic numbers, degree 4.

Formula

Equals (1 + sqrt(13) + sqrt(2*sqrt(13) - 2))/4. - Vaclav Kotesovec, Aug 27 2021

Largest real root of x^4 - x^3 - x^2 - x + 1. - Linas Vepstas, Feb 06 2024

Largest real root of (Sum_{i=0..3} (1/x^i)) ^ 2 - 3 * Sum_{i=0..3} ((1/x^i)^2), from Descartes' theorem. See Comment. - Charles L. Hohn, Nov 13 2025

Example

1.722083805739042245027069212153831462070116555...

Mathematica

z = 2000; r = 9/7;
u = CoefficientList[Series[1/Sum[Round[(k + 1)*r] (-x)^k, {k, 0, z}], {x, 0, z}],
  x];  (* A289916  *)
v = N[u[[z]]/u[[z - 1]], 200]
RealDigits[v, 10][[1]] (* A289917 *)

Prog

(PARI) (1+sqrt(13)+sqrt(2*sqrt(13)-2))/4 \\ Charles R Greathouse IV, Jun 08 2026

Crossrefs

Cf. A078140 (includes guide to related constants), A289916.

Cf. A289915 (spheres about a point), A104457 (circles tangent to a line), A318605 (circles about a point).

Sequence in context: A274570 A334056 A176704 * A198415 A195723 A138996

Adjacent sequences: A289914 A289915 A289916 * A289918 A289919 A289920

Keyword

nonn,cons,easy,changed

Author

Clark Kimberling, Jul 18 2017

Status

approved