# [seqfan] Re: Packing many different square sizes in a square

Allan Wechsler acwacw at gmail.com
Wed Jun 19 17:17:08 CEST 2013

```Thank you, Robert and Giovanni, for the extra hints. Since my last post,
the following obvious lemmas occurred to my slow brain:

1. A(n) is nondecreasing.

2. A tiling of the square of side n is always achievable using squares of
sides 1 .. A(n).

Then it became clear to me that I *should* be asking: what is the side B(k)
of the smallest square into which squares of sides 1 .. k may be packed?
Using the data we have assembled already, I get the following terms for B:

1, 3, 5, 7, 9, 11, 13, 15, 18, 21, 24, 27 ...

And *this* is in OEIS already, the 4-digit classic A005842.  My A(n) is the
largest k with A005842(k) <= n. This sort of "inverse" can be done with any
nondecreasing sequence, so I'm feeling less certain that the new sequence
belongs in OEIS.

Giovanni, I don't know whether it's a typo or a miscalculation, but your
terms are sometimes bigger than mine.  The smallest example is that you
have A(9) = 4, while I have a packing of the 9-square with 5 sizes.  Here
is the tiling:

AAAAABBBB
AAAAABBBB
AAAAABBBB
AAAAABBBB
AAAAACCCC
DDDEECCCC
DDDEECCCC
DDDFFCCCC
GHIFFJKLM
```