A295282 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.

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

Offset: 1

Author

Peter Munn, Nov 19 2017

Keywords

• nonn

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.

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.

Examples

			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

A001622 gives the value of the golden ratio.

Cf. A000045, A007067, A022342, A060645.

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.