JoeBartholomew.com Rectangles, Table 1

The Golden Rectangle and Sequences of Rectangles
with Similar Properties (continued)

Tables	Return
	Table 2. Second Sequence of Rectangles Table 3. Third Sequence of Rectangles

Table 1. First Sequence of Rectangles Based on the Silver Means [7]
	Golden Rectangle	1st Order	2nd Order	3rd Order
Ratio Formula	(a+b)/a = a/b	(a+b)/(a-b) = a/b	(a+b)/(a-2b) = a/b	(a+b)/(a-3b) = a/b
Solution, a/b	(1+√5)/2	1+√2	(3+√13)/2	2+√5
Approximate value	1.61803	2.41421	3.30278	4.23607
Sloane [11]	A000045	A000129	A006190	A001076
Number sequence	F(i) = F(i-1) + F(i-2)	F(i) = 2F(i-1) + F(i-2)	F(i) = 3F(i-1) + F(i-2)	F(i) = 4F(i-1) + F(i-2)
Unit squares to construct rectangle (1)	0.5	1	1.5	2
Squares to remove (2)	1	2	3	4
	4th Order	5th Order	6th Order	7th Order
Ratio Formula	(a+b)/(a-4b) = a/b	(a+b)/(a-5b) = a/b	(a+b)/(a-6b) = a/b	(a+b)/(a-7b) = a/b
Solution, a/b	(5+√29)/2	3+√10	(7+√53)/2	4+√17
Approximate value	5.19258	6.16228	7.14005	8.12311
Sloane [11]	A052918	A005668	A054413	A041025
Number sequence	F(i) = 5F(i-1) + F(i-2)	F(i) = 6F(i-1) + F(i-2)	F(i) = 7F(i-1) + F(i-2)	F(i) = 8F(i-1) + F(i-2)
Unit squares to construct rectangle (1)	2.5	3	3.5	4
Squares to remove (2)	5	6	7	8

(1) Unit squares to construct rectangle provide the length of the long side when the short side is one. The length is the value in the table plus the diagonal from one corner to the opposite corner of the unit squares. See Construction, First Sequence.
(2) Squares to remove are the unit squares to remove when constructing new, progressively smaller rectangles with the same proportions as the first. See Spirals, First Sequence.

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