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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. 1
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 (list; graph; refs; listen; history; text; internal format)
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.

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.

LINKS

Table of n, a(n) for n=1..67.

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

A001622 gives the value of the golden ratio.

Cf. A000045, A007067, A022342, A060645.

Sequence in context: A298863 A184588 A047488 * A066093 A022342 A218606

Adjacent sequences:  A295279 A295280 A295281 * A295283 A295284 A295285

KEYWORD

nonn

AUTHOR

Peter Munn, Nov 19 2017

STATUS

approved

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Last modified October 22 10:57 EDT 2018. Contains 316437 sequences. (Running on oeis4.)