Quaternion "loop" sequence

[Gerald McGarvey]

For this sequence, a(n) - a(n-6) has some patterns, especially at the end:
-8 -4 4 8 4 -4 -8 -4 4 8 4 -4 -8 -4 4 8 4 -4
also sum(a(n)+...+a(n-5)) has this pattern at the end:
-2 -10 -14 -10 -2 2 -2 -10 -14 -10 -2 2 -2 -10 -14 -10 -2 2 -2

At 07:19 PM 6/12/2005, Creighton Dement wrote:
>Hi all,
>
>None of the graphed sequences mentioned in my last posts have been
>submitted (see http://www.crowdog.de/20801/home.html  ) - mainly
>because I first wanted to complete the syllable explanation page
>http://www.crowdog.de/Explanations.html
>and that is taking me a long time (still unsure about the name field-
>perhaps it can be made similar to the one given on this page after the
>rules have all been written down).
>
>In the mean time, I submitted a sequence using only elements from the
>quaternion algebra (over reals) ; the procedure for calculating the
>elements of this sequence is very similar to the other "graphed"
>sequences.  Perhaps the graph isn't quite as spectacular as the others,
>but it seemed simpler for me to explain...  (Maple finds no g.f. after
>200 or so terms). I still have 3 more of these to submit but I wanted to
>wait to see if anyone has recommendations on things I should change,
>first.
>
>  %I A108618
>%S A108618  1, 2, -1, -2, -3, -6, -6, 1, 4, 3, 0, -5, -10, -8, 3, 8, 5,
>-2, -9, -12, -6, 7, 16, 10, -9, -18, -11, 4, 15, 14, -2, -16, -20, -3,
>14, 17, 6, -12, -24, -11, 10, 21, 14, -8, -22, -20, 3, 20, 17, -2, -21,
>-24, -6, 19, 28, 10, -21, -36, -18, 19, 40, 22, -21, -42, -23, 16, 39,
>26, -14, -40, -32, 9, 38, 29, -8, -39, -36, 2, 36, 38, -1, -38, -39, -6,
>32, 42, 7, -34, -43, -14, 28, 46, 15, -30, -47, -22, 24, 50, 23, -26,
>-51
>%N A108618 A quaternion-generated sequence calculated using the rules
>given in the comment box with initial seed x = .5'i + .5'j + .5'k + .5e
>; version: "tes"
>%C A108618 Set y = x = .5'i + .5'j + .5'k + .5e  Define a(0) = 1 (this
>is 2 times the coefficient of the unit e in x), then "loop" steps 1-5,
>below. a(n) is given by 2 times the coefficient of e (the unit) in y
>from step 4 inside of the the n-th loop.
>
>
>Step 1 (Loop 1): Calculate x*y
>Result:  x*y =  .5'i + .5'j + .5'k - .5e
>
>Step 2 (Loop 1): Add the fractional parts of the real coefficienct basis
>vectors of x*y (i.e. 'i, 'j, 'k, e)
>Result: .5 + .5 + .5 - .5 = 1 = s
>
>Step 3 (Loop 1): Calculate x + x*y + se
>Result  .5'i + .5'j + .5'k + .5e + (.5'i + .5'j + .5'k - .5e) + se
>= 'i + 'j + 'k + e.
>
>Step 4 (Loop 1): Set y equal to the result from Step 3.
>Result: y = 'i + 'j + 'k + e ; thus a(1) = 2*1 = 2
>
>Step 5 (Loop 1): Return to Step 1
>
>
>Step 1 (Loop 2):
>Result:  x*y = 'i + 'j + 'k - e
>
>Step 2 (Loop 2):
>Result: s = 0
>
>Step 3 (Loop 2): 1.5'i + 1.5'j + 1.5'k -.5e
>
>Step 4 (Loop 2): y = 1.5'i + 1.5'j + 1.5'k -.5e ; thus, a(2) = 2*(-.5) =
>-1
>
>
>**Loop 1**  + 'i + 'j + 'k + e
>
>**Loop 2**  + 1.5'i + 1.5'j + 1.5'k - .5e
>
>**Loop 3**  + 'i + 'j + 'k - e
>
>**Loop 4**  + .5'i + .5'j + .5'k - 1.5e
>
>**Loop 5**  - 3e
>
>**Loop 6**  - 'i - 'j - 'k - 3e
>
>**Loop 7**  - 1.5'i - 1.5'j - 1.5'k + .5e
>
>**Loop 8**  + 2e
>
>**Loop 9**  + 1.5'i + 1.5'j + 1.5'k + 1.5e
>
>**Loop 10**  + 2'i + 2'j + 2'k
>
>**Loop 11**  + 1.5'i + 1.5'j + 1.5'k - 2.5e
>
>**Loop 12**  - 5e
>%H A108618 C. Dement, <a
>href="http://www.crowdog.de/home.html">The Floretions</a>.
>%H A108618 C. Dement, <a
>href="http://www.crowdog.de/QuaternionLoop.html">Graph of first 500
>terms.</a>.
>%Y A108618 Cf. A108619, A108620, A108621
>%O A108618 0
>%K A108618 ,sign,
>%A A108618 Creighton Dement (crowdog at crowdog.de), Jun 12 2005
>RH
>RA 84.129.245.186
>RU
>RI