A088896 Length of longest integral ladder that can be moved horizontally around the right angled corner where two hallway corridors of integral widths meet.

125, 1000, 2197, 3375, 4913, 8000, 15625, 17576, 24389, 27000, 39304, 42875, 50653, 59319, 64000, 68921, 91125, 125000, 132651, 140608, 148877, 166375, 195112, 216000, 226981, 274625, 314432, 343000, 389017, 405224, 421875, 474552, 512000

Offset

1,1

Comments

The set of values for the integral-widths corridors and longest ladder are merely the cubes of Pythagorean triples, viz. (A046083, A046084, A009000).

The corridors' widths may be parametrically expressed as d*(sin x)^3 and d*(cos x)^3, for a longest ladder length d making an angle x with one of the corridors.

A given ladder, however, is maximum-corner-bending for a family of infinite pairs of perpendicular corridor widths and that the envelope of the maximum bending positions is that of a sliding rod against the outer wall, which is a branch of an astroid or four-cusped hypocycloid.

References

E. Mendelson, 3000 Solved Problems in Calculus, Chapter 16 Problem 16.56 pp. 131, Mc Graw-Hill 1988.

M. Spiegel, Theory and Problems of Advanced Calculus, Chapter 4 Problem 40 pp. 75, Mc Graw-Hill 1974.

Links

Table of n, a(n) for n=1..33.

C. Azeredo, The Ladder Problem

L. Husch and M. Szapiel, The Longest Ladder

M. Kantor, Knox College, Puzzle of the Week

J. J. O'Connor and E. R. Robertson, Astroid

T. Sillke, longest ladder

D. Sjerve, Solution to problem No.3

W. H. Steeb, Solved Problem

Eric Weisstein's World of Mathematics, Astroid

Formula

a(n)=d^3, where d=A009003(n).

Crossrefs

Sequence in context: A204611 A296037 A355885 * A016851 A204795 A352161

Adjacent sequences: A088893 A088894 A088895 * A088897 A088898 A088899

Keyword

nonn

Author

Lekraj Beedassy, Nov 28 2003

Status

approved