[seqfan] The 3d triangulation sequence of the Fibonacci sequence.

[DAN_CYN_J]

Hi all seqfans, 

I entered this once before but not explaining it well. 

Where each term of the Fibonacci sequence starting with (1) has a 
direct correlation in the 3d grid when drawing a line within this grid 
as -- x=1,y=1,z=2,-x=-3,-y=-5,-z=-8,x=13,y=21,z=34, 
-x=55,-y=89,-z=144 and so on. 
Always compleating the 3d cycle of (x,y,z,-x,-y,-z) 
The sequence below is produced from the Fibonacci sequence and is 
used to triangulate the end of the z or -z points giving the distance 
from these end points back to the origin of x=0,y=0,z=0. 
To obtain these distances, starting with the 4th term of this sequence, 
it is just sqrt(x^2+y^2+z^2) and the starting (x) should be where 
n= term number -- n==1(mod 3) 

An example of these triangulated distances and the values taken from 
this sequence are below. 

sqrt(-2^2+-4^2+-6^2) = 7.48331477354788277116.. < f_6 so ceiling of this = 8 
sqrt(11^2+17^2+28^2) =34.55430508634199114328.. > f_9 so floor of this = 34    
sqrt(-44^2+-72^2+-116^2) =143.44336861632886234286.. < f_12 so ceiling of this = 144 
sqrt(189^2+305^2+494^2) =610.55876048092209811936.. > f_15 so floor of this = 610 
and so-on---- where each succeeding calculation the two separate remainders 
converge on themselves. 

Also interesting is the fact that these can be calculated out of order like (y,z,-x) 
sqrt(17^2+28^2+-44^2) = 54.85435260760990878909.. < f_10 so ceiling this = 55 
or --(z,-x,-y) 
sqrt(28^2+-44^2+-72^2) = 88.9044430835714756855.. < f_11 so ceiling this = 89 
Where it appears these out of order calculations need ceiling but 
the next group of out of order need the floor function. 

The sequence used in the triangulation of sqrt(x^2+y^2+z^2) 

 1   1   2  -2  -4  -6   11   17   28  -44  -72  -116   189   305   494  -798 
-1292  -2090   3383   5473   8856  -14328  -23184  -37512   60697   98209 
 158906  -257114  -416020  -673134   1089155   1762289   2851444  -4613732 
-7465176  -12078908   19544085   31622993   51167078  -82790070  -133957148 
-216747218   350704367   567451585   918155952  -1485607536  -2403763488 
-3889371024   6293134513   10182505537   16475640050  -26658145586 
-43133785636  -69791931222   112925716859   182717648081   295643364940 
-478361013020  -774004377960  -1252365390980   2026369768941   3278735159921 
 5305104928862  -8583840088782  -13888945017644  -22472785106426 
 36361730124071   58834515230497   95196245354568  -154030760585064 
-249227005939632  -403257766524696 

The input to produce this sequence is the Fibonacci sequence. 

The primes that produce another sequence in this 3d triangulation is also interesting. 

Cheers, 

Dan