Consecutive Rectangles in Similar Rectangles

The following pictures show similar rectangles with short sides 1, 2, 3, ... n with aspect ratio r packed inside another similar rectangle with short side s with the largest known proportion p of coverage.

r = any
s = 1
p = 1
Trivial.

r = 1 + √2
s = 1 + √2
p = 15 – 10 √2 = .857+
Found by Maurizio Morandi in June 2015.

r = √15 / 3
s = √15
p = 14/15 = .933+
Found by Maurizio Morandi in June 2015.

4.

r = (3 + √73) / 8
s = (3 + √73) / 2
p = 15 (3√73 – 41) / 256 = .900+
Found by Maurizio Morandi in June 2015. 5.

r = 2 √15 / 5
s = 2 √15
p = 11/12 = .916+
Found by Maurizio Morandi in June 2015. 6.

r = (√273 – 3) / 12
s = (3 + √273) / 2
p = 91 (47 – √273) / 2904 = .955+
Found by Maurizio Morandi in June 2015.

7.

r = 2
s = 12
p = 35/36 = .972+
Found by Maurizio Morandi in June 2015. 8.

r = √210 / 14
s = √210
p = 34/35 = .971+
Found by Maurizio Morandi in June 2015. 9.

r = (15 + √365) / 14
s = (15 + √365) / 2
p = 57 (59 – 3 √365) / 98 = .980+
Found by Maurizio Morandi in June 2015.

10.

r = √391 / 17
s = √391
p = 385/391 = .984+
Found by Maurizio Morandi in June 2015. 11.

r = 3 √57 / 19
s = 3 √57
p = 506/513 = .986+
Found by Maurizio Morandi in June 2015. 12.

r = √165 / 11
s = 2 √165
p = 65/66 = .984+
Found by Maurizio Morandi in June 2015.

13.

r = 5 / √19
s = 7 + 5 √19
p = 91 (262 – 35 √19) / 10082 = .987+
Found by Maurizio Morandi in June 2015. 14.

r = 3/2
s = 32
p = 1015/1024 = .991+
Found by Maurizio Morandi in June 2015. 15.

r = 3 √66 / 11 – 1
s = 11 + 3 √66
p = 1240 (65 – 6 √66) / 20339 = .991+
Found by Maurizio Morandi in June 2015.

16.

r = 2 √42 / 9
s = 6 √42
p = 187/189 = .989+
Found by Maurizio Morandi in June 2015. 17.

r = √1794 / 39
s = √1794
p = 595/598 = .994+
Found by Maurizio Morandi in June 2015. 18.

r = √2135 / 35
s = √2135
p = 2109/2135 = .987+
Found by Maurizio Morandi in June 2015.