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Consider $a(n)$, the minimal number of squares into which the oblong of size $(n+1)\times(n)$ can be divided. What is the behavior of $a(n)$? The first 379 terms of the oblong square packing sequence A279317 are known. We can guess that the largest oblongs divisible into 1, 2, 3, ... squares are at positions 1, 2, 3, 6, 10, 18, 34, 55, 104, 176, 307, 551, 969, 1698, 2925, 5210, 8730, 15130, 25678, 49582, with the first 11 verified. Here are what those oblongs look like:
maximal oblongs For 551 and up, they are from the Catalogues of Simple Perfect Squared Rectangles, and it's purely a wild guess that maximal dissections beyond a certain size won't repeat a square size. The following image of best dissections for oblongs 349 to 387 (from Filling rectangles with integer-sided squares) shows that square values are repeated frequently. The second one, oblong 349, is first oblong requiring 14 squares. The minimal oblongs needing 2, 3, 4, ... squares are at {1, 2, 3, 4, 9, 7, 16, 19, 44, 69, 113, 179, 349}. What are the first oblongs needing 15, 16, 17, 18 squares?

oblongs of size 349 to 387

Consider $b(n) = round(n^{1/3}) +6$. The rounded cube root seems to be tightly bound to $a(n)$. Here's a plot of $b(n)-a(n)$. This is sequence A321028.

oblong plot

$b(18)-a(18) = 2$. All $b(n)-a(n)$ terms from there to $n$=387 are -1, 0, 1. We can calculate $b(49582)-a(49582) = 22$. Is that an anomaly in general behavior, or do $b(n)$ and $a(n)$ diverge at some point?

In addition to other sources above, much of this data is at Minimally Squared Rectangles. It was inspired by the Mondrian Upper Bound problem $-$ I wondered if other packing problems might have weird bounds. Oblong dissections are closely related to the Mrs. Perkins's Quilt problem.

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  • $\begingroup$ Do you have data for arbitrary n X m rectangles? $\endgroup$ – Peter Kagey Dec 13 '16 at 19:37
  • $\begingroup$ Peter, see the third or fourth link for data on arbitrary rectangles with max side 380 $\endgroup$ – Ed Pegg Dec 14 '16 at 0:35

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