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

DAN_CYN_J dan_cyn_j at comcast.net
Mon Oct 7 13:55:09 CEST 2013

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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

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