An example of a periodic floretion sequence.
Creighton Dement
crowdog at crowdog.de
Sun Feb 6 01:03:48 CET 2005
Dear Seqfans,
There is probably not any reason at the moment to submit a bunch of
sequences of the type (period 61) below- perhaps a single example will
suffice. However, I consider what is happening to be interesting. In
addition, periodic sequences appear to play a role in "force
transformations" (my main topic of interest at the moment)- though I
shall not go into that now.
Ex. floretion: + .5'i + .5'j + .5'k + .5e
1vesfizsumrokseq: [[5]], -3, -6, -5, 3, 2, -1, -1, 2, 3, 7, -6, -13, -3,
6, 9, -7, -10, 9, 5, 2, -11, -15, 6, 3, 7, -14, -17, 11, 2, -5, -13,
-10, 13, -3, -2, -15, -15, 6, -3, -7, -10, -3, 11, -6, -9, -9, -6, -1,
-5, -2, -5, -1, 2, -3, -7, -2, 1, -3, -2, [[5]], -3, -6, -5, 3, 2, -1,
-1, 2, 3, 7, -6, -13, -3, 6, 9, -7, -10, 9, 5, 2
The term [[5]] in brackets denotes where the sequence begins to repeat
itself.
Here is a very brief (and incomplete!) explanation of what each of the
syllables in the word "1vesfizsumrokseq" means:
1 - the sequence was not multiplied by any factor other than 1 (if the
floretion had, instead, generated a sequence of fractions which, for
ex., turned out to be half of the above sequence, then FAMP would have
recognized it and automatically written 2vesfizsumrokseq: 5, -3, -6, -5,
...)
ves - sum up over all basis vector coefficients (as previously
explained).
For ex., **0** starts at + 'i + 'j + 'k + 2e (see below) and
ves(+ 'i + 'j + 'k + 2e) = 1 + 1 + 1 + 2 = 5. This is the first term of
the sequence.
ves(- 2.5j' - .5'jj' + 2.5'ij' - 2.5'kj' ) is the second term, etc.
fiz - roughly: multiply the floretion resulting from each stage by a
single floretion basis vector by one of the quasi-pentagrams featured at
http://www.crowdog.de/13829/19001.html and do this cyclicly.
Ex. http://www.research.att.com/projects/OEIS?Anum=A101348
sum - roughly: add the floretion from the previous stage to the result
of the next stage. **0** starts at + 'i + 'j + 'k + 2e because the
original (generating) floretion was + .5'i + .5'j + .5'k + .5e and
this added to itelf is 'i + 'j + 'k + e. Why is the coefficient of the
unit basis vector "e" only 1 and not 2 ? The reason is that the "rok"
symmetry is also present:
rok - roughly: (note: there are several versions of "fiz", "sum" and
"rok". For ex., the factorial type sequences - including double
factorial sequences - were an alternate version of rok) multiply all
basis vectors of the given floretion together (in any order as signs
will be disregarded) which have nonzero coefficients. In the case of,
say, 2'i + 14'j we would have 'i * 'j = 'k. Next, add the resulting
basis vector (i.e. 'k) to the original floretion 2'i + 14'j (yielding
2'i + 14'j + 'k) and use this result when calculating.
seq - the sequence generated
Note: in a previous posting, I mentioned a few period 24 sequences and
commented something to the effect "I don't know why this is happening
and probably never will". That was a similar procedure... I find it
difficult to understand what is happening "exactly" because
"1vesfizsumrokseq" combines everything listed above into a single step
in order to get to the next term of the sequence. That said, observe
how these steps alter the floretion at each stage until it reaches the
60th term.
**0** + 'i + 'j + 'k + 2e
**1** - 2.5j' - .5'jj' + 2.5'ij' - 2.5'kj'
**2** + 'i - 4'j - 'k - j' - 'jj' + 'kj' - e
**3** + .5i' - 2.5k' + 1.5'ii' - 1.5'kk' - .5'ik' + .5'ji' - 2.5'jk' -
.5'ki'
**4** - 'j + 'k - .5i' + k' - .5'ii' - 2'kk' + 'ik' + .5'ji' + 3'jk'
+ 1.5'ki' - e
**5** + 1.5'i - .5'j + .5'k - i' - .5k' + .5'kk' - 1.5'ik' - 'ji' +
3.5'jk' + 'ki' - .5e
**6** - .5i' - j' - 1.5k' + 2.5'ii' - 'jj' + .5'kk' + .5'ik' - 1.5'ji'
+ .5'jk' + 2.5'ki' - 2'kj'
**7** - 'i - 2'j + .5i' - j' + 3k' - 1.5'ii' - 'jj' + 'kk' + 2'ik' +
.5'ji' - 'jk' - .5'ki' + 'kj' - e
**8** - 1.5'i + 3.5'j + .5'k + .5i' + j' + 1.5'ii' + 'jj' - 'kk' -
'ik' + .5'ji' - 'jk' - .5'ki' - 'kj' - .5e
**9** - 1.5'i - 3.5'j + 1.5'k - .5i' + .5j' + .5k' - .5'ii' - .5'jj' -
.5'kk' + .5'ij' + 1.5'ik' + .5'ji' + .5'jk' + 1.5'ki' + 1.5'kj' + 1.5e
**10** + 3.5'i - 1.5'j + 1.5'k - i' - .5j' - .5k' - .5'jj' - .5'kk' +
1.5'ij' + .5'ik' - 'ji' + 1.5'jk' + 'ki' + .5'kj' + 2.5e
**11** + 'i + 'j + 'k - i' - 2.5j' - 1.5k' - .5'jj' + .5'kk' + .5'ij'
+ .5'ik' - 'ji' + .5'jk' + 'ki' - 5.5'kj'
**12** - 1.5'i - 4.5'j - 1.5'k + .5i' - 1.5j' + .5k' - 1.5'ii' -
1.5'jj' + 1.5'kk' - 1.5'ij' + .5'ik' + .5'ji' - .5'jk' - .5'ki' +
1.5'kj' - 3.5e
**13** + .5'i + 1.5'j - .5'k + j' - 3k' + 3'ii' + 'jj' - 2'kk' - 3'ik'
- 'kj' - .5e
**14** - 'i - 'j + 2'k - 1.5i' + .5j' + 4k' - 1.5'ii' - .5'jj' - 2'kk'
+ .5'ij' + 'ik' + 1.5'ji' + 'jk' + 1.5'ki' + 1.5'kj'
**15** + 2'i - 'j + 2'k - 1.5i' - .5j' + 2k' - 1.5'ii' - .5'jj' + 'kk'
+ 1.5'ij' + 'ik' - 1.5'ji' + 4'jk' + 1.5'ki' + .5'kj'
**16** + 'i + 'j + 'k - 3i' - .5j' - 3k' + 3'ii' - 1.5'jj' + .5'ij' -
2'ji' - 3.5'kj'
**17** - 2'i - 3'j - 'k + 1.5i' - 1.5j' + k' - 1.5'ii' - 1.5'jj' +
'kk' - 1.5'ij' + 2'ik' + 1.5'ji' - 4'jk' - 1.5'ki' + 1.5'kj' - e
**18** + 'i + 4'j + 'k + 1.5j' - .5k' + 3'ii' + 1.5'jj' - 2.5'kk' +
1.5'ij' - 1.5'ik' - .5'jk' - 1.5'kj' + 2e
**19** - 2'i - 4'j + 4'k - 1.5i' + 2k' - 1.5'ii' + 2'ik' + 1.5'ji' +
'jk' + 1.5'ki' + 3'kj' - e
**20** + 3.5'i - 4.5'j + 1.5'k - 1.5i' - 1.5j' - .5k' - 1.5'ii' -
1.5'jj' + .5'kk' + 1.5'ij' + 1.5'ik' - 1.5'ji' + 2.5'jk' + 1.5'ki' +
1.5'kj' + .5e
**21** + 1.5'i + 1.5'j + 1.5'k - 1.5i' - 3k' + .5'ii' - 2'ij' -
2.5'ji' + .5'ki' - 6'kj' - 1.5e
**22** - 4'i - 2'j - 2'k + 1.5i' - j' + k' - 1.5'ii' - 'jj' + 2'kk' -
3'ij' + 1.5'ji' - 2'jk' - 1.5'ki' + 'kj' - 4e
**23** + 1.5'i + 2.5'j + .5'k - i' + 1.5j' - k' + 3'ii' + 1.5'jj' -
2'kk' + 1.5'ij' - 3'ik' - 'ji' + 3'jk' + 'ki' - 1.5'kj' - .5e
**24** - 2.5'i - 1.5'j + 2.5'k - 2i' + 4.5k' - 2'ii' + .5'kk' + .5'ik'
+ 2'ji' - 1.5'jk' + 3'kj' - .5e
**25** + 1.5'i - 2.5'j + 2.5'k - i' - 1.5j' + 2.5k' - 3'ii' - 1.5'jj'
+ 1.5'kk' + 1.5'ij' + 3.5'ik' - 'ji' + 1.5'jk' + 'ki' + 1.5'kj' + .5e
**26** + 1.5'i + 1.5'j + 1.5'k - 3i' + .5j' - 3k' + 'ii' - .5'jj' -
'kk' + .5'ij' - 'ik' - 2'ji' - 'jk' - 3'ki' - 4.5'kj' - 1.5e
**27** - 2.5'i - 2.5'j - 2.5'k + 2i' - j' - 1.5k' - 'jj' + .5'kk' -
3'ij' - .5'ik' + 2'ji' - 4.5'jk' - 2'ki' + 'kj' - 1.5e
**28** + 3.5'i + 1.5'j + 1.5'k - i' + j' - k' + 3'ii' + 'jj' - 3'kk' +
3'ij' - 'ik' - 'ji' + 'jk' + 'ki' - 'kj' + 2.5e
**29** - 2'i - 2'j + 4'k - 2i' - j' + 3k' - 2'ii' + 'jj' + 'kk' - 'ij'
+ 'ik' + 2'ji' + 'jk' + 3'kj' - 4e
**30** + 'i - 5'j + 'k - i' - 2j' - 3'ii' - 2'jj' + 2'kk' + 2'ik' -
'ji' + 2'jk' + 'ki' + 2'kj' - 2e
**31** + 'i + 'j + 'k - i' + 2.5j' - 3k' + 'ii' + .5'jj' - 'kk' -
2.5'ij' - 'ik' - 3'ji' - 'jk' - 'ki' - 3.5'kj' - 3e
**32** - 4'i + 'j - 2'k + 2i' + k' + 'kk' - 3'ij' - 'ik' + 2'ji' -
3'jk' - 2'ki' - 2e
**33** + 2'i + 2'j + 2'k - 1.5i' + j' + 1.5k' + 1.5'ii' + 'jj' -
1.5'kk' + 3'ij' - .5'ik' - 1.5'ji' + 3.5'jk' + 1.5'ki' - 'kj'
**34** - 3'i - 2'j + 2'k - 1.5i' - j' + 2k' - 1.5'ii' + 'jj' + 3'kk' -
'ij' + 1.5'ji' - 2'jk' - 1.5'ki' + 3'kj' - 2e
**35** + .5'i - 3.5'j + 1.5'k - 2j' + .5k' - 3'ii' - 2'jj' + 1.5'kk' +
3.5'ik' - 1.5'jk' + 2'kj' + .5e
**36** + 'i + 'j + 'k - .5i' + j' - 1.5k' - 1.5'ii' + 'jj' - 1.5'kk' -
1.5'ik' - 1.5'ji' - 1.5'jk' - 3.5'ki' - 4'kj' - 3e
**37** - 2'i - 'j - 3'k + 1.5i' - 2k' + 1.5'ii' - 3'ij' - 3'ik' +
1.5'ji' - 2'jk' - 1.5'ki' - 2e
**38** + 3.5'i - 1.5'j + 1.5'k - 1.5i' - k' + 1.5'ii' - 2'kk' + 3'ij'
- 1.5'ji' + 2'jk' + 1.5'ki' + .5e
**39** - 1.5'i + .5'j + 1.5'k - 1.5i' - 1.5j' + 2.5k' - 1.5'ii' +
1.5'jj' + 1.5'kk' - 1.5'ij' - .5'ik' + 1.5'ji' + .5'jk' - 1.5'ki' +
1.5'kj' - 4.5e
**40** - 1.5'i - 2.5'j + .5'k - 1.5j' + .5k' - 3'ii' - 1.5'jj' +
2.5'kk' - 1.5'ij' + 1.5'ik' + .5'jk' + 1.5'kj' - 2.5e
**41** + 2.5j' - 1.5k' + 'ii' + .5'jj' - 1.5'kk' - .5'ij' - 1.5'ik' -
2'ji' - 1.5'jk' - 2'ki' - .5'kj' - 3e
**42** - 1.5'i + 1.5'j - 1.5'k + 1.5i' + .5j' + .5k' + 1.5'ii' +
.5'jj' - .5'kk' - 1.5'ij' - 1.5'ik' + 1.5'ji' - 2.5'jk' - 1.5'ki' -
.5'kj' + .5e
**43** + 1.5'i + .5'j + 2.5'k - i' + 2k' - 'kk' + 3'ij' + 2'ik' - 'ji'
+ 'jk' + 'ki' + .5e
**44** - 2'i - 2'j + 'k - .5i' - 1.5j' - k' - .5'ii' + 1.5'jj' + 3'kk'
- 1.5'ij' + .5'ji' - 1.5'ki' + 1.5'kj' - 3e
**45** - 3'j + .5i' - 1.5j' - 2k' - 1.5'ii' - 1.5'jj' + 'kk' - 1.5'ij'
+ 'ik' + .5'ji' - 2'jk' - .5'ki' + 1.5'kj'
**46** + 2i' + .5j' - 2'ii' + 1.5'jj' - 'kk' - .5'ij' - 'ik' - 'ji' -
'jk' - 'ki' - 2.5'kj' - 3e
**47** - 'i - 2'k + .5i' + .5j' + 1.5'ii' + .5'jj' - 1.5'ij' - 3'ik' +
.5'ji' + 'jk' - .5'ki' - .5'kj' - 2e
**48** + 'i - 2'j + 'k - i' - .5j' - .5k' - .5'jj' - .5'kk' + 1.5'ij'
+ .5'ik' - 'ji' + 1.5'jk' + 'ki' + .5'kj' - 2e
**49** - 'i + 'j - 'k - .5i' - j' + k' - .5'ii' + 'jj' + 'kk' - 'ij' -
'ik' + .5'ji' - 1.5'ki' - 2e
**50** - 1.5'i + .5'j + .5'k + .5i' - .5j' + .5k' - 1.5'ii' - .5'jj' +
1.5'kk' - 1.5'ij' + .5'ik' + .5'ji' - .5'jk' - .5'ki' + .5'kj' - .5e
**51** - .5'i - .5'j - .5'k + .5i' + .5'ii' - 'kk' + 2'ij' - 'ik' -
.5'ji' - 'jk' - 1.5'ki' - 1.5e
**52** + 'i - 'j - 'k + .5i' + 1.5'ii' - 'kk' - 'ik' + .5'ji' - 'jk' -
.5'ki' + e
**53** + .5'i - .5'j + 1.5'k - .5j' - .5'jj' - 'kk' + 1.5'ij' + 2'ik'
- 2'jk' + .5'kj' + .5e
**54** - .5'i - 1.5'j + .5'k - j' - 1.5k' + 'jj' + .5'kk' - 'ij' +
.5'ik' + 2.5'jk' - 2.5e
**55** + .5'i - 1.5'j - .5'k - .5j' - 2.5k' - .5'jj' + .5'kk' -
1.5'ij' - 1.5'ik' + .5'jk' + .5'kj' - .5e
**56** - .5'i - .5'j - .5'k + 2i' - .5j' + .5'jj' - .5'ij' - 'ji' +
2'ki' - 1.5'kj' - 1.5e
**57** - .5'i - .5'j - .5'k + 2.5k' + .5'kk' - .5'ik' + 1.5'jk' - 1.5e
**58** - 1.5'i + .5'j + .5'k - 2.5e
**59** - 'i - 'j - 'k + e
**60** + 'i + 'j + 'k + 2e
Sincerely,
Creighton
-- Sonja is bigger than me and I am bigger than Sonja! (my 3-year-old
daughter expressing that she and her friend Sonja are exactly the same
height)
More information about the SeqFan
mailing list