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A295282
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a(n) > n is chosen to minimize the difference between ratios a(n):n and n:(a(n) - n), so that they are matching approximations to the golden ratio.
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1
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2, 3, 5, 7, 8, 10, 11, 13, 15, 16, 18, 19, 21, 23, 24, 26, 28, 29, 31, 32, 34, 36, 37, 39, 40, 42, 44, 45, 47, 49, 50, 52, 53, 55, 57, 58, 60, 61, 63, 65, 66, 68, 70, 71, 73, 74, 76, 78, 79, 81, 83, 84, 86, 87, 89, 91, 92, 94, 95, 97, 99, 100, 102, 104, 105, 107, 108
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OFFSET
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1,1
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COMMENTS
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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.
More specifically, for k > 0: F(3k) is an even Fibonacci number, F(3k+1) is odd and a(F(3k)/2) = F(3k+1)/2 + 1/2; whereas A007067(F(6k+3)/2) = F(6k+4)/2 + 1/2, but A007067(F(6k)/2) = F(6k+1)/2 - 1/2.
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LINKS
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FORMULA
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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.
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EXAMPLE
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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...
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CROSSREFS
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A001622 gives the value of the golden ratio.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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