|
%I #12 Dec 19 2017 02:46:50
%S 2,3,5,7,8,10,11,13,15,16,18,19,21,23,24,26,28,29,31,32,34,36,37,39,
%T 40,42,44,45,47,49,50,52,53,55,57,58,60,61,63,65,66,68,70,71,73,74,76,
%U 78,79,81,83,84,86,87,89,91,92,94,95,97,99,100,102,104,105,107,108
%N 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.
%C The difference between the matching ratios is evaluated by dividing the larger by the smaller.
%C 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.
%C 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.
%C 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.
%C 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.
%F a(n) = (m+n) > n so as to minimize (max((m+n)/n, n/m) / min((m+n)/n, n/m)).
%F 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.
%e The matching ratios and the differences between them begin:
%e 2:1 1:1 2.0
%e 3:2 2:1 1.3333...
%e 5:3 3:2 1.1111...
%e 7:4 4:3 1.3125
%e 8:5 5:3 1.0416...
%e 10:6 6:4 1.1111...
%e 11:7 7:4 1.1136...
%e 13:8 8:5 1.015625
%e 15:9 9:6 1.1111...
%e 16:10 10:6 1.0416...
%e 18:11 11:7 1.0413...
%e 19:12 12:7 1.0827...
%e 21:13 13:8 1.0059...
%e 23:14 14:9 1.0561...
%e 24:15 15:9 1.0416...
%e 26:16 16:10 1.015625
%e 28:17 17:11 1.0657...
%e 29:18 18:11 1.0156...
%e 31:19 19:12 1.0304...
%e 32:20 20:12 1.0416...
%e 34:21 21:13 1.0022...
%e ...
%e For n = 4:
%e 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;
%e if a(4) = 6, ratios would be 6:4 and 4:2, with difference = (4/2) / (6/4) = 16/12 = 1.333...;
%e if a(4) = 7, ratios would be 7:4 and 4:3, with difference = (7/4) / (4/3) = 21/16 = 1.3125;
%e if a(4) = 8, ratios would be 8:4 and 4:4, with difference = (8/4) / (4/4) = 32/16 = 2.0.
%e 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.
%e This example translates as follows into the geometry described early in the comments:
%e n 4 4
%e m 2 3
%e n+m 6 7
%e Rectangle A n X (n+m) 4 X 6 4 X 7
%e Rectangle B m X n 2 X 4 3 X 4
%e Scaling ratio n:(n+m) 4:6 4:7
%e m scaled up m*(n+m)/n 2*6/4 3*7/4
%e = side of C l 3 5.25
%e Rectangle C l X (n+m) 3 X 6 5.25 X 7
%e Rectangle D (n+l) X (n+m) 7 X 6 9.25 X 7
%e proportion C/D l/(n+l) 3/7 5.25/9.25
%e - as decimal 0.4285... 0.5675...
%e - its difference from 0.5 0.0714... 0.0675...
%Y A001622 gives the value of the golden ratio.
%Y Cf. A000045, A007067, A022342, A060645.
%K nonn
%O 1,1
%A _Peter Munn_, Nov 19 2017
|
