Various results: Molien series of Q_8; binomial transform of Pell

[creigh at o2online.de]

Hi,

So much time has been spent updating FAMP over the past few weeks that all new 
sequence submissions by me have been put on the backburner. 

Three results which (the reader should always take my ignorance into account) I 
see as potentially interesting:

1st:    A094038(n+2) = A088013(n+1) + A007070(n) 

A088013, Binomial transform of Chebyshev polynomials evauluated at 3
A094038,  Binomial transform of (Pell(-n)+Pell(n))/2  -or-  Binomial tranform 
of Pell(n)(1-(-1)^n)/2
A007070, Preceded by 0, this is the binomial transform of the Pell numbers 
A000129 

Above relation generated by:
- 0.5'i + 0.5'k - 0.5i' + 0.5k' - 1'jj' + 0.5'ij' + 1'ik' - 0.5'ji' - 0.5'jk' 
+ 1'ki' + 0.5'kj' + 1e 



2nd result:  "achu[I]+tes", below, apparently corresponds to A028242
( 2achu[K]+tesseq also reverses adjacent numbers)

A028242, Follow n by n-1. Also (essentially) Molien series of 
2-dimensional quaternion group Q_8

(Chu's group, subgroup symmetries:) 
  
 achu[J]+tesseq: -2, 8, -13, 23, -32, 46, -59, 77, -94, 116, -137, 163, -188, 
218, -247, 281 
 2achu[K]+tesseq: -1, 6, -5, 10, -9, 14, -13, 18, -17, 22, -21, 26, -25, 
30, -29, 34 
 achu[I]+tesseq: 1, 0, 2, -1, 3, -2, 4, -3, 5, -4, 6, -5, 7, -6, 8, -7  

Above relation generated by:
 - 0.5'j - 0.5'k - 0.5j' - 0.5k' + 1'jj' + 0.5'kk' - 0.5'ik' - 0.5'ki' 
- 0.5e 

One may also note that  achu[J]+tesseq(2n+1) correspsond to the (nice) sequence 
A033951. 
A033951, Write 1,2,... in clockwise spiral; sequence gives numbers on positive 
x axis.


 
3rd result:   2'j + 0.5'k + 2i' - 4j' + 0.5k' - 0.5'ii' - 0.5'jj' + 2'kk' + 3'ik' 
+ 1'jk' - 3'ki' - 1'kj'  generates the sequence
(3,1,-1,1,3,1,-1,1,3,1,-1,1,3,1,-1,1,..) = gcd(Fib(n+5), Fib(n+1))
(see below). However, it could have been just a "coincidence". 

 (Emmy's Three, subgroup symmetries:) 
  
 em[I]seq: 3, 1, -1, 1, 3, 1, -1, 1, 3, 1, -1, 1, 3, 1, -1, 1 
 em[I*]seq: -1, -8, 9, 0, -1, -8, 9, 0, -1, -8, 9, 0, -1, -8, 9, 0
 
 em[J]seq: 2, -7, 8, 1, 2, -7, 8, 1, 2, -7, 8, 1, 2, -7, 8, 1 
 em[J*]seq: 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
 
 em[K]seq: -1, 1, 3, 1, -1, 1, 3, 1, -1, 1, 3, 1, -1, 1, 3, 1 
 em[K*]seq: 3, -8, 5, 0, 3, -8, 5, 0, 3, -8, 5, 0, 3, -8, 5, 0
 
 famseq: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
 fam*seq: 1, -8, 7, 0, 1, -8, 7, 0, 1, -8, 7, 0, 1, -8, 7, 0 


Sincerely, 
Creighton http://www.crowdog.de/13829/home.html