Crystal Ball sequences and Polytope numbers via force transforms.

Sun Mar 6 00:45:31 CET 2005

Hello Seqfans, 

I just began looking at 
H. K. Kim, "On Regular polytope numbers", 
http://com2mac.postech.ac.kr/papers/2001/01-22.pdf
so the "conjecture" I speak of, below, might not actually be one at all.
However, I think it is a nice example nevertheless. 

A direct application of the identity ves = jes + les + tes appears to 
lead to a conjecture involving Crystal ball sequences and polytope
numbers:  

Floretion: + .5'i + .5i' + .5'jk' + .5'kj' + e ( = I*J )
I =  + .5'i + .5'kj' + e
J =  + .5i' + .5'jk' + e
ForType: 1A
LoopType: tes. 

0-th iteration ************

1vesseq: 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35,
37,  

 1tesseq: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1,  

 1lesseq: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18,
19, 20,  

 1jesseq: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18,
19, 20,  

1st iteration **********

1vesforseq: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1,  

 1tesforseq: -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1,
1, -1, 1, -1 

 1lesforseq: 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0,
1, 0, 1, 0 

 1jesforseq: 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0,
1, 0, 1, 0 

2nd iteration ************

1vesforseq: -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1,
1, -1, 1, -1 

 1tesforseq: -3, 5, -7, 9, -11, 13, -15, 17, -19, 21, -23, 25, -27, 29,
-31, 33 

 1lesforseq: 1, -2, 3, -4, 5, -6, 7, -8, 9, -10, 11, -12, 13, -14, 15,
-16, 17,  

 1jesforseq: 1, -2, 3, -4, 5, -6, 7, -8, 9, -10, 11, -12, 13, -14, 15,
-16, 17,  

3rd iteration **********

1vesforseq: -3, 5, -7, 9, -11, 13, -15, 17, -19, 21, -23, 25, -27, 29,
-31, 33 

 1tesforseq: -5, 13, -25, 41, -61, 85, -113, 145, -181, 221, -265, 313,
-365,  

 1lesforseq: 1, -4, 9, -16, 25, -36, 49, -64, 81, -100, 121, -144, 169,
-196,  

 1jesforseq: 1, -4, 9, -16, 25, -36, 49, -64, 81, -100, 121, -144, 169,
-196,  

4th iteration **********

1vesforseq: -5, 13, -25, 41, -61, 85, -113, 145, -181, 221, -265, 313,
-365,  
 Centered square numbers: 2n(n+1)+1.

 1tesforseq: -7, 25, -63, 129, -231, 377, -575, 833, -1159, 1561, -2047 

 1lesforseq: 1, -6, 19, -44, 85, -146, 231, -344, 489, -670, 891, -1156,

 1jesforseq: 1, -6, 19, -44, 85, -146, 231, -344, 489, -670, 891, -1156 

Octahedral numbers: (2n^3 + n)/3.
http://www.research.att.com/projects/OEIS?Anum=A005900

5th iteration *************

1vesforseq: -7, 25, -63, 129, -231, 377, -575, 833, -1159, 1561, -2047, 

 Centered octahedral numbers (crystal ball sequence for cubic
 lattice)

 1tesforseq: -9, 41, -129, 321, -681, 1289, -2241, 3649, -5641, 8361,
-11969, 16641, -22569, 29961, -39041, 50049, -63241, 78889, -97281,  
Centered 4-dimensional orthoplex numbers (crystal ball sequence for
4-dimensional cubic lattice)

 1lesforseq: 1, -8, 33, -96, 225, -456, 833, -1408, 2241, -3400, 4961, -
,  
 4-cross polytope 

 1jesforseq: 1, -8, 33, -96, 225, -456, 833, -1408, 2241, -3400, 4961, -

4-cross polytope 

6th iteration **********************

1vesforseq: -9, 41, -129, 321, -681, 1289, -2241, 3649, -5641, 8361,  
 Crystal ball sequence for 4-dimensional cubic lattice

 1tesforseq: -11, 61, -231, 681, -1683, 3653, -7183, 13073, -22363,  
 Crystal ball sequence for 5-dimensional cubic lattice.

 1lesforseq: 1, -10, 51, -180, 501, -1182, 2471, -4712, 8361, -14002 
 The  5-dimensional cross-polytope, which is represented by the Schlafli
 symbol {3, 3, 3, 4}. It is the dual of the 5-dimensional hypercube.
5-cross polytope 

1jesforseq: 1, -10, 51, -180, 501, -1182, 2471, -4712, 8361, -14002,  
5-cross polytope 

7th iteration *********************

1vesforseq: -11, 61, -231, 681, -1683, 3653, -7183, 13073, -22363,  
Crystal ball sequence for 5-dimensional cubic lattice. 

 1tesforseq: -13, 85, -377, 1289, -3653, 8989, -19825, 40081, -75517,  
 Crystal ball sequence for 6-dimensional cubic lattice.

 1lesforseq: 1, -12, 73, -304, 985, -2668, 6321, -13504, 26577, -48940, 

 6-dimensional  hyperoctahedron, which is represented by the Schlafli
symbol {3, 3, 3, 3, 4}. It is the dual of the 6-dimensional hypercube.
6-cross polytope 

 1jesforseq: 1, -12, 73, -304, 985, -2668, 6321, -13504, 26577, -48940, 

6-cross polytope 

Note: (just for comparison) taken from http://www.crowdog.de/forceII.txt

Floretion:  .5j' + .5'kk' + .5'ki' + .5e 
ForType: 1A
LoopType: jes (or les), after transforming the constant sequence 
(1, 1, 1, 1, 1, 1,). 

Conjecture: the m-th iteration of lesfor/jesfor returns lesfor/jesfor
sequences equal to "Binomial coefficients binomial(n,m)", apart from
initial terms. [The sequence tesfor then relates binomial(n,m) to
binomial(n,m-1) via the identity ves = jes + les + tes]. In other words,
the jesfor-trasform of the sequence binomial(n,m-1) under the floretion
.5j' + .5'kk' + .5'ki' + .5e  is the sequence "Binomial coefficients
binomial(n,m)". 

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)