[seqfan] simple zero-divisors with Cayley-Dickson algebras

[Joerg Arndt]

Write A for the element (as a vector) where all
components are zero but component A is +1.
The most simple zero-divisors we have are
those of the form (A + B).
If (A+B) is a ZD, then all 4 of ( +-A +-B ) are ZDs.

Simple ZD are indeed both left and right ZD
(so always proper ZD).


Counting the ZD of the form (A+B) gives

  n:  0, 1, 2, 3,   4,   5,   6,     7,     8,
 ZD:  0, 0, 0, 0,  42,  294, 1518,  6942,  29886,

(note the number of all simple ZD is 4 times this one,
e.g. there are 168 simple ZD for the sedenions).

I conjecture the sequence is the same as the following:

? v=vector(14); v[4]=42;
? for(k=5,#v, v[k]=2*v[k-1]+(2^(k-1)-1)*(2^(k-1)-2) ); v
 [0, 0, 0, 42, 294, 1518, 6942, 29886, 124542, 509694, 2064894, 8317950, 33400830, 133885950]

(this is via observing a very striking structure).

OEIS worthy?