OFFSET
1,1
COMMENTS
The difference between the matching ratios is evaluated by dividing the larger by the smaller.
Take a rectangle A with sides n and n+m; remove a square of side n from one end to form rectangle B with sides n and m; scale B in the ratio (n+m):n to form rectangle C with a side n+m. Place A and C alongside, with edges of length n+m coinciding, to form rectangle D. For n > 0, let m_n be the m that has the coincident edges dividing D in nearest to equal proportions, then a(n) = n + m_n.
Compared with other neighboring values of n, the resulting proportions can be made most nearly equal when n is a Fibonacci number F(k) = A000045(k), k > 1, in which case a(n) is F(k+1). In contrast, if 2n is a Fibonacci number F(k), then a relatively good choice for rectangle A's longer side would be F(k+1)/2, except that F(k+1) is odd when F(k) is even, so F(k+1)/2 is halfway between integers.
a(n) is usually the same as A007067(n), but when 2n is a Fibonacci number, they sometimes differ. The first differences are a(4) = 7 = A007067(4) + 1 and a(72) = 117 = A007067(72) + 1. The author expects a(n) to differ from A007067(n) if and only if n is in A060645. The terms of A060645 are half the value of alternate even Fibonacci numbers.
FORMULA
a(n) = (m+n) > n so as to minimize (max((m+n)/n, n/m) / min((m+n)/n, n/m)).
a(n+1) = a(n) + 2 - floor((a(n)+2) * (a(n)+1-n) * (a(n)+1) * (a(n)-n) / (n+1)^4), with a(0) = 0 for the purpose of this calculation.
EXAMPLE
The matching ratios and the differences between them begin:
2:1 1:1 2.0
3:2 2:1 1.3333...
5:3 3:2 1.1111...
7:4 4:3 1.3125
8:5 5:3 1.0416...
10:6 6:4 1.1111...
11:7 7:4 1.1136...
13:8 8:5 1.015625
15:9 9:6 1.1111...
16:10 10:6 1.0416...
18:11 11:7 1.0413...
19:12 12:7 1.0827...
21:13 13:8 1.0059...
23:14 14:9 1.0561...
24:15 15:9 1.0416...
26:16 16:10 1.015625
28:17 17:11 1.0657...
29:18 18:11 1.0156...
31:19 19:12 1.0304...
32:20 20:12 1.0416...
34:21 21:13 1.0022...
...
For n = 4:
if a(4) = 5, the matching ratios would be a(4):4 = 5:4 and 4:(a(4)-4) = 4:1, with the difference between them (larger divided by smaller) = (4/1) / (5/4) = 16/5 = 3.2;
if a(4) = 6, ratios would be 6:4 and 4:2, with difference = (4/2) / (6/4) = 16/12 = 1.333...;
if a(4) = 7, ratios would be 7:4 and 4:3, with difference = (7/4) / (4/3) = 21/16 = 1.3125;
if a(4) = 8, ratios would be 8:4 and 4:4, with difference = (8/4) / (4/4) = 32/16 = 2.0.
Any larger value for a(4) would give a difference between the ratios that exceeded 2.0, so a(4) = 7, as this achieves the minimum difference.
This example translates as follows into the geometry described early in the comments:
n 4 4
m 2 3
n+m 6 7
Rectangle A n X (n+m) 4 X 6 4 X 7
Rectangle B m X n 2 X 4 3 X 4
Scaling ratio n:(n+m) 4:6 4:7
m scaled up m*(n+m)/n 2*6/4 3*7/4
= side of C l 3 5.25
Rectangle C l X (n+m) 3 X 6 5.25 X 7
Rectangle D (n+l) X (n+m) 7 X 6 9.25 X 7
proportion C/D l/(n+l) 3/7 5.25/9.25
- as decimal 0.4285... 0.5675...
- its difference from 0.5 0.0714... 0.0675...
CROSSREFS
KEYWORD
nonn
AUTHOR
Peter Munn, Nov 19 2017
STATUS
approved