A134799 a(n) = 3^((3^n - 1)/2).

1, 3, 81, 1594323, 12157665459056928801, 5391030899743293631239539488528815119194426882613553319203

Offset: 0

Author

Yasutoshi Kohmoto, Jan 09 2008

Keywords

• nonn
• easy

Comments

Number of partitions into "bus routes" of the graph G_{n+1} defined below.

These seem to be one-third the reduced denominators of Newton's iteration for 1/sqrt(3), starting with 1/3. - Steven Finch, Oct 08 2024

Examples

			.........|..................G_1
****
.......__|__................G_2
.........|
****
.__|_____|_____|__..........G_3
...|.....|.....|
.........|
.......__|__
.........|
****.
..._|_........._|_..........G_4
_|__|_____|_____|__|_
.|._|_....|...._|_.|
....|.....|.....|
......_|__|__|_
.......|._|_.|
..........|
****
G_1 = o---. = rooted tree with one edge and one leaf node. For n > 0, G_{n+1} is obtained from G_n by splitting each leaf node into three.
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..7
• Author?, Mitsumata tree
• X. Gourdon and P. Sebah, Pythagoras' Constant.

Crossrefs

Cf. A131709, A020760, A376870.

Programs

Mathematica

3^((3^Range[0, 6] - 1)/2) (* Paolo Xausa, Oct 17 2024 *)

Formula

a(n) is conjectured to be one-third the reduced denominator of b(n) = (3/2)*b(n-1)*(1 - b(n-1)^2); b(0) = 1/3. - Steven Finch, Oct 08 2024

Limit_{n -> oo} A376870(n)/(3*a(n)) = 1/sqrt(3) = A020760. - Steven Finch, Oct 08 2024

Extensions

Edited by N. J. A. Sloane, Jan 29 2008

a(5) from Andrew Howroyd, Oct 07 2024