Spiro-Fibonacci differences

Wed Jan 8 02:20:12 CET 2003

Spiro-Fibonacci differences.

Write terms in square boxes, working in a spiral, cf. Ulam prime spiral,
such that a(n) = difference of the two previous terms written closest to
the box for a(n). a(0)=0 and a(1)=1. (I follow the Fibonacci convention
by starting the indices with 0).

The sequence of Spiro-Fibonacci differences runs:

0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0,
1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0,
0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0,
0, 0, 0, 1, 0, 1

As the spiral fills squares of increasing size, the number of 1s
increases as follows. b(n) = the sum of the first n^2 differences, i.e.
the number of 1s when the spiral fills a square of height n and width n.

0, 3, 7, 11, 15, 21, 28, 35, 45, 55, 68, 84, 96, 111, 125, 140, 154,
172, 192, 212, 236, 254, 273, 291, 311, 330, 359, 392, 420

The frequency of 1s as n increases is:

n       frequency = b(n)/n^2
1       0
2       0.75
3       0.777777778
4       0.6875
5       0.6
6       0.583333333
7       0.571428571
8       0.546875
9       0.555555556
10      0.55
11      0.561983471
12      0.583333333
13      0.568047337
14      0.566326531
15      0.555555556
16      0.546875
17      0.532871972
18      0.530864198
19      0.531855956
20      0.53
21      0.535147392
22      0.524793388
23      0.516068053
24      0.505208333
25      0.4976
26      0.48816568
27      0.492455418
28      0.5
29      0.49940547
...     ...
50      0.4176

Not sure how the frequency behaves as n grows very large. Here are the
numbers of 1s in the nth 'L' as an (n-1) x (n-1) square grows into an n
x n square:

n       no. of 1s in nth L
1       0
2       3
3       4
4       4
5       4
6       6
7       7
8       7
9       10
10      10
11      13
12      16
13      12
14      15
15      14
16      15
17      14
18      18
19      20
20      20
21      24
22      18
23      19
24      18
25      20
26      19
27      29
28      33
29      28

1s can be coloured in on large squares in the same way as Ulam coloured
in primes. The following shows the 29 x 29 square:

[ Section: 1/1 File: Spiro-Fibonacci_differences.gif UUencoded by: Turnpike Integrated Version 5.01 S ]

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end

sum -r/size 28296/9014 section (from "begin" to "end")
sum -r/size 45722/6506 entire input file

And the 50 x 50 square:

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`
end

sum -r/size 35259/7565 section (from "begin" to "end")
sum -r/size 42450/5463 entire input file

Given the simple constraints (square packing, difference the influence
from two closest squares), I wonder if this occurs physically.

Neil

-- 
Neil Fernandez