# Inverse of 12th cyclotomic polynomial + update

creigh at o2online.de creigh at o2online.de
Tue Nov 9 22:41:55 CET 2004

```The even/odd symmetry update to FAMP has been
http://www.crowdog.de/13829/home.html
The newer version also includes an even/odd+sigma symmetry which
additionally applies the "swap operator" (normally written with a sigma).

An example- observe the element  .5 'k + .5 'kk' + .5 'ki' + .5 'kj' . As
discussed in the Floretion paper, it can be shown that the above element
is one member of a group isomorph to Q x C_3 where Q are the quaternions.

(The sequences window of) FAMP now returns:
4vesseqsig: 8, 0, 8, 0, 0, 0, -8, 0, -8, 0, 0, 0, 8, 0, 8, 0, 0, 0, -8,
0, -8, 0, 0, 0, 8, 0, 8, 0, 0, 0, -8
divided by 8, this is the signed sequence
http://www.research.att.com/projects/OEIS?Anum=A014021
Inverse of 12th cyclotomic polynomial

4tesseqsig: 0, 1, 1, 0, 1, -1, 0, -1, -1, 0, -1, 1, 0, 1, 1, 0, 1, -1, 0,
-1, -1, 0, -1, 1, 0, 1, 1, 0, 1, -1, 0
http://www.research.att.com/projects/OEIS?Anum=A010892
Inverse of 6th cyclotomic polynomial.

4lesseqsig: 6, 1, 3, 2, -3, 1, -6, -1, -3, -2, 3, -1, 6, 1, 3, 2, -3, 1,
-6, -1, -3, -2, 3, -1, 6, 1, 3, 2, -3, 1, -6  //not listed
2jesseqsig: 1, -1, 2, -1, 1, 0, -1, 1, -2, 1, -1, 0, 1, -1, 2, -1, 1,
0, -1, 1, -2, 1, -1, 0, 1, -1, 2, -1, 1, 0, -1 //not listed

identity: vessig = jessig + lessig + tessig

posseqsig: 2, 2, 3, 1, 2, 2, 0, 2, 1, 1, 2, 2, 2, 2, 3, 1, 2, 2, 0, 2, 1,
1, 2, 2, 2, 2, 3, 1, 2, 2, 0
nesseqsig: 0, -2, -1, -1, -2, -2, -2, -2, -3, -1, -2, -2, 0, -2, -1, -1,
-2, -2, -2, -2, -3, -1, -2, -2, 0, -2, -1, -1, -2, -2, -2

identity: possig + neqsig = vessig

Furthermore, the sequence given this morning now appears in the sequences window as:

evenseq: 0, 1, 1, 0, 0, 5, 8, 13, 21, 0, 0, 89, 144, 233, 377,,
oddseq: 0, 0, 0, 2, 3, 0, 0, 0, 0, 34, 55, 0, 0, 0, 0, 610, 987,

even + odd = ves // = fib

Lastly, the element
+ 0.25'i + 0.5'j - 0.75'k - 0.25i' - 0.25k' + 0.5'ii' + 0.25'ij' + 0.75'ji'
- 0.25'jk' + 0.5'ki' + 0.25'kj' + 0.5e

gives us

2evenseqsig: 4, 2, 4, 6, 4, 5, 2, 2, 10, 11, 20, 22, 12, 13, 2, 2, 18, 19,
36, 38, 20, 21, 2, 2, 26, 27, 52, 54, 28, 29, 2
2oddseqsig: 0, 0, 0, 0, 4, 5, 10, 12, 6, 7, 0, 0, 12, 13, 26, 28, 14,
15, 0, 0, 20, 21, 42, 44, 22, 23, 0, 0, 28, 29, 58

vesseqsig: 2, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18,
19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30

evensig + oddsig = vessig // (apart from initial term) = natural numbers
where each block of nonnegative integers from oddseqsig can be read off
as:
2. double result ( =2*(n+1) )
3. add 2 to result (= 2*(n+1) + 2)
4. half result (=n+2)
5. add 1 to result (=n+3)

Thanks very much for your time.

Sincerely,
Creighton

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