A233699 Ideal rectangle side length for packing squares with side 1/n.

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

Offset: 0

Author

John W. Nicholson, Dec 15 2013

Keywords

• nonn
• cons

Comments

With one side s_1 = 1/2+1/3 = 5/6, and with area A = s_1*s_2 = sum(n=2,infinity, 1/n^2) = Pi^2/6 - 1 = A013661 - 1, the second side, s_2, can be solved.

The current packing record holder is Marc Paulhus, who developed a packing algorithm (see Link).

Examples

			0.77392088021787172376689819997523022706273988144815812528266987524400896448...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000
• M. M. Paulhus, An Algorithm for Packing Squares, Journal of Combinatorial Theory,1998, A,82(2), pages 147-157.
• Pegg Jr, Ed., Wolfram Demonstrations Project, Packing Squares with Side 1/n
• Wikipedia, Packing Squares with Side 1/n

Crossrefs

Essentially the same as A164102.

Programs

Magma

C := ComplexField(); (Pi(C)^2-6)/5 // G. C. Greubel, Jan 26 2018

Mathematica

RealDigits[(Pi^2-6)/5,10,120][[1]] (* Harvey P. Dale, Aug 21 2017 *)

PARI

(Pi^2-6)/5;

Formula

Equals (Pi^2-6)/5 = A164102/10 - 6/5.