JoeBartholomew.com Rectangles, Table 3

The Golden Rectangle — One Rectangle in a Sequence of
Infinitely Many Rectangles (continued)

Tables	Return
	Table 1. First Sequence of Rectangles Table 2. Second Sequence of Rectangles

Table 3. Third Sequence of Rectangles
		1st Order	2nd Order	3rd Order
Ratio Formula	b/(a–2*b/1) = a/b	b/(a–2*b/3) = a/b	b/(a–2*b/5) = a/b	b/(a–2*b/7) = a/b
Solution, a/b	1+√2	(1+√10)/3	(1+√26)/5	(1+√50)/7
Approximate value	2.41421	1.38743	1.21980	1.15301
Number sequence	F(i) = 2F(i-1) + F(i-2)	F(i) = 2F(i-1)/3 + F(i-2)	F(i) = 2F(i-1)/5 + F(i-2)	F(i) = 2F(i-1)/7 + F(i-2)
Unit squares to construct rectangle (1)	1	1/3	1/5	1/7
Squares to remove (2)	2	2/3	2/5	2/7
	4th Order	5th Order	6th Order	7th Order
Ratio Formula	b/(a–2*b/9) = a/b	b/(a–2*b/11) = a/b	b/(a–2*b/13) = a/b	b/(a–2*b/15) = a/b
Solution, a/b	(1+√82)/9	(1+√122)/11	(1+√170)/13	(1+√226)/15
Approximate value	1.11727	1.09503	1.07988	1.06889
Number sequence	F(i) = 2F(i-1)/9 + F(i-2)	F(i) = 2F(i-1)/11 + F(i-2)	F(i) = 2F(i-1)/13 + F(i-2)	F(i) = 2F(i-1)/15 + F(i-2)
Unit squares to construct rectangle (1)	1/9	1/11	1/13	1/15
Squares to remove (2)	2/9	2/11	2/13	2/15

(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.
(2) Squares to remove are the fractions of a unit square to remove when constructing new, progressively smaller rectangles with the same proportions as the first.

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