SPECIAL ROOTS FOR F_4 LIE ALGREBRA :      

 

As in the paper math-ph/0302061, our notation here is

R[A]={j1, j2, j3, j4}

replaces

R[A]={gam[1,j1], gam[2,j2], gam[3,j3], gam[4,j4]} .

for A=1,2, ... ,1152.

where

a[i]'s are simple roots of F_4.

 

**********************************************************************

gam[1, 1]:=0

gam[1, 2]:=a[1]

gam[1, 3]:=a[1] + a[2]

gam[1, 4]:=a[1] + a[2] + 2 a[3]

gam[1, 5]:=a[1] + a[2] + 2 a[3] + 2 a[4]

gam[1, 6]:=a[1] + 2 a[2] + 2 a[3]

gam[1, 7]:=a[1] + 2 a[2] + 2 a[3] + 2 a[4]

gam[1, 8]:=a[1] + 2 a[2] + 4 a[3] + 2 a[4]

gam[1, 9]:=a[1] + 3 a[2] + 4 a[3] + 2 a[4]

gam[1,10]:=2 a[1] + 2 a[2] + 2 a[3]

gam[1,11]:=2 a[1] + 2 a[2] + 2 a[3] + 2 a[4]

gam[1,12]:=2 a[1] + 2 a[2] + 4 a[3] + 2 a[4]

gam[1,13]:=2 a[1] + 4 a[2] + 4 a[3] + 2 a[4]

gam[1,14]:=2 a[1] + 4 a[2] + 6 a[3] + 2 a[4]

gam[1,15]:=2 a[1] + 4 a[2] + 6 a[3] + 4 a[4]

gam[1,16]:=3 a[1] + 3 a[2] + 4 a[3] + 2 a[4]

gam[1,17]:=3 a[1] + 4 a[2] + 4 a[3] + 2 a[4]

gam[1,18]:=3 a[1] + 4 a[2] + 6 a[3] + 2 a[4]

gam[1,19]:=3 a[1] + 4 a[2] + 6 a[3] + 4 a[4]

gam[1,20]:=3 a[1] + 5 a[2] + 6 a[3] + 2 a[4]

gam[1,21]:=3 a[1] + 5 a[2] + 6 a[3] + 4 a[4]

gam[1,22]:=3 a[1] + 5 a[2] + 8 a[3] + 4 a[4]

gam[1,23]:=3 a[1] + 6 a[2] + 8 a[3] + 4 a[4]

gam[1,24]:=4 a[1] + 6 a[2] + 8 a[3] + 4 a[4]

 

gam[2, 1]:=0

gam[2, 2]:=a[2]

gam[2, 3]:=a[2] + 2 a[3]

gam[2, 4]:=a[2] + 2 a[3] + 2 a[4]

gam[2, 5]:=2 a[2] + 2 a[3]

gam[2, 6]:=2 a[2] + 2 a[3] + 2 a[4]

gam[2, 7]:=2 a[2] + 4 a[3] + 2 a[4]

gam[2, 8]:=3 a[2] + 4 a[3] + 2 a[4]

gam[2, 9]:=a[1] + a[2]

gam[2,10]:=a[1] + a[2] + 2 a[3]

gam[2,11]:=a[1] + a[2] + 2 a[3] + 2 a[4]

gam[2,12]:=a[1] + 3 a[2] + 2 a[3]

gam[2,13]:=a[1] + 3 a[2] + 2 a[3] + 2 a[4]

gam[2,14]:=a[1] + 3 a[2] + 4 a[3]

gam[2,15]:=a[1] + 3 a[2] + 4 a[3] + 4 a[4]

gam[2,16]:=a[1] + 3 a[2] + 6 a[3] + 2 a[4]

gam[2,17]:=a[1] + 3 a[2] + 6 a[3] + 4 a[4]

gam[2,18]:=a[1] + 5 a[2] + 6 a[3] + 2 a[4]

gam[2,19]:=a[1] + 5 a[2] + 6 a[3] + 4 a[4]

gam[2,20]:=a[1] + 5 a[2] + 8 a[3] + 4 a[4]

gam[2,21]:=2 a[1] + 2 a[2] + 2 a[3]

gam[2,22]:=2 a[1] + 2 a[2] + 2 a[3] + 2 a[4]

gam[2,23]:=2 a[1] + 2 a[2] + 4 a[3] + 2 a[4]

gam[2,24]:=2 a[1] + 3 a[2] + 2 a[3]

gam[2,25]:=2 a[1] + 3 a[2] + 2 a[3] + 2 a[4]

gam[2,26]:=2 a[1] + 3 a[2] + 4 a[3]

gam[2,27]:=2 a[1] + 3 a[2] + 4 a[3] + 4 a[4]

gam[2,28]:=2 a[1] + 3 a[2] + 6 a[3] + 2 a[4]

gam[2,29]:=2 a[1] + 3 a[2] + 6 a[3] + 4 a[4]

gam[2,30]:=2 a[1] + 4 a[2] + 4 a[3]

gam[2,31]:=2 a[1] + 4 a[2] + 4 a[3] + 4 a[4]

gam[2,32]:=2 a[1] + 4 a[2] + 8 a[3] + 4 a[4]

gam[2,33]:=2 a[1] + 5 a[2] + 4 a[3] + 2 a[4]

gam[2,34]:=2 a[1] + 5 a[2] + 8 a[3] + 2 a[4]

gam[2,35]:=2 a[1] + 5 a[2] + 8 a[3] + 6 a[4]

gam[2,36]:=2 a[1] + 6 a[2] + 6 a[3] + 2 a[4]

gam[2,37]:=2 a[1] + 6 a[2] + 6 a[3] + 4 a[4]

gam[2,38]:=2 a[1] + 6 a[2] + 8 a[3] + 2 a[4]

gam[2,39]:=2 a[1] + 6 a[2] + 8 a[3] + 6 a[4]

gam[2,40]:=2 a[1] + 6 a[2] + 10 a[3] + 4 a[4]

gam[2,41]:=2 a[1] + 6 a[2] + 10 a[3] + 6 a[4]

gam[2,42]:=2 a[1] + 7 a[2] + 8 a[3] + 4 a[4]

gam[2,43]:=2 a[1] + 7 a[2] + 10 a[3] + 4 a[4]

gam[2,44]:=2 a[1] + 7 a[2] + 10 a[3] + 6 a[4]

gam[2,45]:=3 a[1] + 3 a[2] + 4 a[3] + 2 a[4]

gam[2,46]:=3 a[1] + 5 a[2] + 4 a[3] + 2 a[4]

gam[2,47]:=3 a[1] + 5 a[2] + 8 a[3] + 2 a[4]

gam[2,48]:=3 a[1] + 5 a[2] + 8 a[3] + 6 a[4]

gam[2,49]:=3 a[1] + 7 a[2] + 8 a[3] + 2 a[4]

gam[2,50]:=3 a[1] + 7 a[2] + 8 a[3] + 6 a[4]

gam[2,51]:=3 a[1] + 7 a[2] + 12 a[3] + 6 a[4]

gam[2,52]:=3 a[1] + 9 a[2] + 12 a[3] + 6 a[4]

gam[2,53]:=4 a[1] + 5 a[2] + 6 a[3] + 2 a[4]

gam[2,54]:=4 a[1] + 5 a[2] + 6 a[3] + 4 a[4]

gam[2,55]:=4 a[1] + 5 a[2] + 8 a[3] + 4 a[4]

gam[2,56]:=4 a[1] + 6 a[2] + 6 a[3] + 2 a[4]

gam[2,57]:=4 a[1] + 6 a[2] + 6 a[3] + 4 a[4]

gam[2,58]:=4 a[1] + 6 a[2] + 8 a[3] + 2 a[4]

gam[2,59]:=4 a[1] + 6 a[2] + 8 a[3] + 6 a[4]

gam[2,60]:=4 a[1] + 6 a[2] + 10 a[3] + 4 a[4]

gam[2,61]:=4 a[1] + 6 a[2] + 10 a[3] + 6 a[4]

gam[2,62]:=4 a[1] + 7 a[2] + 8 a[3] + 2 a[4]

gam[2,63]:=4 a[1] + 7 a[2] + 8 a[3] + 6 a[4]

gam[2,64]:=4 a[1] + 7 a[2] + 12 a[3] + 6 a[4]

gam[2,65]:=4 a[1] + 8 a[2] + 8 a[3] + 4 a[4]

gam[2,66]:=4 a[1] + 8 a[2] + 12 a[3] + 4 a[4]

gam[2,67]:=4 a[1] + 8 a[2] + 12 a[3] + 8 a[4]

gam[2,68]:=4 a[1] + 9 a[2] + 10 a[3] + 4 a[4]

gam[2,69]:=4 a[1] + 9 a[2] + 10 a[3] + 6 a[4]

gam[2,70]:=4 a[1] + 9 a[2] + 12 a[3] + 4 a[4]

gam[2,71]:=4 a[1] + 9 a[2] + 12 a[3] + 8 a[4]

gam[2,72]:=4 a[1] + 9 a[2] + 14 a[3] + 6 a[4]

gam[2,73]:=4 a[1] + 9 a[2] + 14 a[3] + 8 a[4]

gam[2,74]:=4 a[1] + 10 a[2] + 12 a[3] + 6 a[4]

gam[2,75]:=4 a[1] + 10 a[2] + 14 a[3] + 6 a[4]

gam[2,76]:=4 a[1] + 10 a[2] + 14 a[3] + 8 a[4]

gam[2,77]:=5 a[1] + 7 a[2] + 8 a[3] + 4 a[4]

gam[2,78]:=5 a[1] + 7 a[2] + 10 a[3] + 4 a[4]

gam[2,79]:=5 a[1] + 7 a[2] + 10 a[3] + 6 a[4]

gam[2,80]:=5 a[1] + 9 a[2] + 10 a[3] + 4 a[4]

gam[2,81]:=5 a[1] + 9 a[2] + 10 a[3] + 6 a[4]

gam[2,82]:=5 a[1] + 9 a[2] + 12 a[3] + 4 a[4]

gam[2,83]:=5 a[1] + 9 a[2] + 12 a[3] + 8 a[4]

gam[2,84]:=5 a[1] + 9 a[2] + 14 a[3] + 6 a[4]

gam[2,85]:=5 a[1] + 9 a[2] + 14 a[3] + 8 a[4]

gam[2,86]:=5 a[1] + 11 a[2] + 14 a[3] + 6 a[4]

gam[2,87]:=5 a[1] + 11 a[2] + 14 a[3] + 8 a[4]

gam[2,88]:=5 a[1] + 11 a[2] + 16 a[3] + 8 a[4]

gam[2,89]:=6 a[1] + 9 a[2] + 12 a[3] + 6 a[4]

gam[2,90]:=6 a[1] + 10 a[2] + 12 a[3] + 6 a[4]

gam[2,91]:=6 a[1] + 10 a[2] + 14 a[3] + 6 a[4]

gam[2,92]:=6 a[1] + 10 a[2] + 14 a[3] + 8 a[4]

gam[2,93]:=6 a[1] + 11 a[2] + 14 a[3] + 6 a[4]

gam[2,94]:=6 a[1] + 11 a[2] + 14 a[3] + 8 a[4]

gam[2,95]:=6 a[1] + 11 a[2] + 16 a[3] + 8 a[4]

gam[2,96]:=6 a[1] + 12 a[2] + 16 a[3] + 8 a[4]

 

gam[3, 1]:=0

gam[3, 2]:=a[3]

gam[3, 3]:=a[3] + a[4]

gam[3, 4]:=a[2] + a[3]

gam[3, 5]:=a[2] + a[3] + a[4]

gam[3, 6]:=a[2] + 2 a[3]

gam[3, 7]:=a[2] + 2 a[3] + 2 a[4]

gam[3, 8]:=a[2] + 3 a[3] + a[4]

gam[3, 9]:=a[2] + 3 a[3] + 2 a[4]

gam[3,10]:=2 a[2] + 3 a[3] + a[4]

gam[3,11]:=2 a[2] + 3 a[3] + 2 a[4]

gam[3,12]:=2 a[2] + 4 a[3] + 2 a[4]

gam[3,13]:=a[1] + a[2] + a[3]

gam[3,14]:=a[1] + a[2] + a[3] + a[4]

gam[3,15]:=a[1] + a[2] + 2 a[3]

gam[3,16]:=a[1] + a[2] + 2 a[3] + 2 a[4]

gam[3,17]:=a[1] + a[2] + 3 a[3] + a[4]

gam[3,18]:=a[1] + a[2] + 3 a[3] + 2 a[4]

gam[3,19]:=a[1] + 2 a[2] + 2 a[3]

gam[3,20]:=a[1] + 2 a[2] + 2 a[3] + 2 a[4]

gam[3,21]:=a[1] + 2 a[2] + 3 a[3]

gam[3,22]:=a[1] + 2 a[2] + 3 a[3] + 3 a[4]

gam[3,23]:=a[1] + 2 a[2] + 5 a[3] + 2 a[4]

gam[3,24]:=a[1] + 2 a[2] + 5 a[3] + 3 a[4]

gam[3,25]:=a[1] + 3 a[2] + 3 a[3] + a[4]

gam[3,26]:=a[1] + 3 a[2] + 3 a[3] + 2 a[4]

gam[3,27]:=a[1] + 3 a[2] + 5 a[3] + a[4]

gam[3,28]:=a[1] + 3 a[2] + 5 a[3] + 4 a[4]

gam[3,29]:=a[1] + 3 a[2] + 6 a[3] + 2 a[4]

gam[3,30]:=a[1] + 3 a[2] + 6 a[3] + 4 a[4]

gam[3,31]:=a[1] + 4 a[2] + 5 a[3] + 2 a[4]

gam[3,32]:=a[1] + 4 a[2] + 5 a[3] + 3 a[4]

gam[3,33]:=a[1] + 4 a[2] + 6 a[3] + 2 a[4]

gam[3,34]:=a[1] + 4 a[2] + 6 a[3] + 4 a[4]

gam[3,35]:=a[1] + 4 a[2] + 7 a[3] + 3 a[4]

gam[3,36]:=a[1] + 4 a[2] + 7 a[3] + 4 a[4]

gam[3,37]:=2 a[1] + 2 a[2] + 3 a[3] + a[4]

gam[3,38]:=2 a[1] + 2 a[2] + 3 a[3] + 2 a[4]

gam[3,39]:=2 a[1] + 2 a[2] + 4 a[3] + 2 a[4]

gam[3,40]:=2 a[1] + 3 a[2] + 3 a[3] + a[4]

gam[3,41]:=2 a[1] + 3 a[2] + 3 a[3] + 2 a[4]

gam[3,42]:=2 a[1] + 3 a[2] + 5 a[3] + a[4]

gam[3,43]:=2 a[1] + 3 a[2] + 5 a[3] + 4 a[4]

gam[3,44]:=2 a[1] + 3 a[2] + 6 a[3] + 2 a[4]

gam[3,45]:=2 a[1] + 3 a[2] + 6 a[3] + 4 a[4]

gam[3,46]:=2 a[1] + 4 a[2] + 4 a[3] + 2 a[4]

gam[3,47]:=2 a[1] + 4 a[2] + 5 a[3] + a[4]

gam[3,48]:=2 a[1] + 4 a[2] + 5 a[3] + 4 a[4]

gam[3,49]:=2 a[1] + 4 a[2] + 7 a[3] + 2 a[4]

gam[3,50]:=2 a[1] + 4 a[2] + 7 a[3] + 5 a[4]

gam[3,51]:=2 a[1] + 4 a[2] + 8 a[3] + 4 a[4]

gam[3,52]:=2 a[1] + 5 a[2] + 6 a[3] + 2 a[4]

gam[3,53]:=2 a[1] + 5 a[2] + 6 a[3] + 4 a[4]

gam[3,54]:=2 a[1] + 5 a[2] + 7 a[3] + 2 a[4]

gam[3,55]:=2 a[1] + 5 a[2] + 7 a[3] + 5 a[4]

gam[3,56]:=2 a[1] + 5 a[2] + 9 a[3] + 4 a[4]

gam[3,57]:=2 a[1] + 5 a[2] + 9 a[3] + 5 a[4]

gam[3,58]:=2 a[1] + 6 a[2] + 8 a[3] + 4 a[4]

gam[3,59]:=2 a[1] + 6 a[2] + 9 a[3] + 4 a[4]

gam[3,60]:=2 a[1] + 6 a[2] + 9 a[3] + 5 a[4]

gam[3,61]:=3 a[1] + 4 a[2] + 5 a[3] + 2 a[4]

gam[3,62]:=3 a[1] + 4 a[2] + 5 a[3] + 3 a[4]

gam[3,63]:=3 a[1] + 4 a[2] + 6 a[3] + 2 a[4]

gam[3,64]:=3 a[1] + 4 a[2] + 6 a[3] + 4 a[4]

gam[3,65]:=3 a[1] + 4 a[2] + 7 a[3] + 3 a[4]

gam[3,66]:=3 a[1] + 4 a[2] + 7 a[3] + 4 a[4]

gam[3,67]:=3 a[1] + 5 a[2] + 6 a[3] + 2 a[4]

gam[3,68]:=3 a[1] + 5 a[2] + 6 a[3] + 4 a[4]

gam[3,69]:=3 a[1] + 5 a[2] + 7 a[3] + 2 a[4]

gam[3,70]:=3 a[1] + 5 a[2] + 7 a[3] + 5 a[4]

gam[3,71]:=3 a[1] + 5 a[2] + 9 a[3] + 4 a[4]

gam[3,72]:=3 a[1] + 5 a[2] + 9 a[3] + 5 a[4]

gam[3,73]:=3 a[1] + 6 a[2] + 7 a[3] + 3 a[4]

gam[3,74]:=3 a[1] + 6 a[2] + 7 a[3] + 4 a[4]

gam[3,75]:=3 a[1] + 6 a[2] + 9 a[3] + 3 a[4]

gam[3,76]:=3 a[1] + 6 a[2] + 9 a[3] + 6 a[4]

gam[3,77]:=3 a[1] + 6 a[2] + 10 a[3] + 4 a[4]

gam[3,78]:=3 a[1] + 6 a[2] + 10 a[3] + 6 a[4]

gam[3,79]:=3 a[1] + 7 a[2] + 9 a[3] + 4 a[4]

gam[3,80]:=3 a[1] + 7 a[2] + 9 a[3] + 5 a[4]

gam[3,81]:=3 a[1] + 7 a[2] + 10 a[3] + 4 a[4]

gam[3,82]:=3 a[1] + 7 a[2] + 10 a[3] + 6 a[4]

gam[3,83]:=3 a[1] + 7 a[2] + 11 a[3] + 5 a[4]

gam[3,84]:=3 a[1] + 7 a[2] + 11 a[3] + 6 a[4]

gam[3,85]:=4 a[1] + 6 a[2] + 8 a[3] + 4 a[4]

gam[3,86]:=4 a[1] + 6 a[2] + 9 a[3] + 4 a[4]

gam[3,87]:=4 a[1] + 6 a[2] + 9 a[3] + 5 a[4]

gam[3,88]:=4 a[1] + 7 a[2] + 9 a[3] + 4 a[4]

gam[3,89]:=4 a[1] + 7 a[2] + 9 a[3] + 5 a[4]

gam[3,90]:=4 a[1] + 7 a[2] + 10 a[3] + 4 a[4]

gam[3,91]:=4 a[1] + 7 a[2] + 10 a[3] + 6 a[4]

gam[3,92]:=4 a[1] + 7 a[2] + 11 a[3] + 5 a[4]

gam[3,93]:=4 a[1] + 7 a[2] + 11 a[3] + 6 a[4]

gam[3,94]:=4 a[1] + 8 a[2] + 11 a[3] + 5 a[4]

gam[3,95]:=4 a[1] + 8 a[2] + 11 a[3] + 6 a[4]

gam[3,96]:=4 a[1] + 8 a[2] + 12 a[3] + 6 a[4]

 

gam[4, 1]:=0

gam[4, 2]:=a[4]

gam[4, 3]:=a[3] + a[4]

gam[4, 4]:=a[2] + a[3] + a[4]

gam[4, 5]:=a[2] + 2 a[3] + a[4]

gam[4, 6]:=a[2] + 2 a[3] + 2 a[4]

gam[4, 7]:=a[1] + a[2] + a[3] + a[4]

gam[4, 8]:=a[1] + a[2] + 2 a[3] + a[4]

gam[4, 9]:=a[1] + a[2] + 2 a[3] + 2 a[4]

gam[4,10]:=a[1] + 2 a[2] + 2 a[3] + a[4]

gam[4,11]:=a[1] + 2 a[2] + 2 a[3] + 2 a[4]

gam[4,12]:=a[1] + 2 a[2] + 3 a[3] + a[4]

gam[4,13]:=a[1] + 2 a[2] + 3 a[3] + 3 a[4]

gam[4,14]:=a[1] + 2 a[2] + 4 a[3] + 2 a[4]

gam[4,15]:=a[1] + 2 a[2] + 4 a[3] + 3 a[4]

gam[4,16]:=a[1] + 3 a[2] + 4 a[3] + 2 a[4]

gam[4,17]:=a[1] + 3 a[2] + 4 a[3] + 3 a[4]

gam[4,18]:=a[1] + 3 a[2] + 5 a[3] + 3 a[4]

gam[4,19]:=2 a[1] + 3 a[2] + 4 a[3] + 2 a[4]

gam[4,20]:=2 a[1] + 3 a[2] + 4 a[3] + 3 a[4]

gam[4,21]:=2 a[1] + 3 a[2] + 5 a[3] + 3 a[4]

gam[4,22]:=2 a[1] + 4 a[2] + 5 a[3] + 3 a[4]

gam[4,23]:=2 a[1] + 4 a[2] + 6 a[3] + 3 a[4]

gam[4,24]:=2 a[1] + 4 a[2] + 6 a[3] + 4 a[4]

 

*******************************************************************************

 

R[ 1]={1, 1, 1,1}

R[ 2]={1, 1, 1,2}

R[ 3]={1, 1, 2,1}

R[ 4]={1, 1, 2,3}

R[ 5]={1, 1, 3,2}

R[ 6]={1, 1, 3,3}

R[ 7]={1, 2, 1,1}

R[ 8]={1, 2, 1,2}

R[ 9]={1, 2, 4,1}

R[ 10]={1, 2, 4,4}

R[ 11]={1, 2, 5,2}

R[ 12]={1, 2, 5,4}

R[ 13]={1, 3, 2,1}

R[ 14]={1, 3, 2,3}

R[ 15]={1, 3, 6,1}

R[ 16]={1, 3, 6,5}

R[ 17]={1, 3, 8,3}

R[ 18]={1, 3, 8,5}

R[ 19]={1, 4, 3,2}

R[ 20]={1, 4, 3,3}

R[ 21]={1, 4, 7,2}

R[ 22]={1, 4, 7,6}

R[ 23]={1, 4, 9,3}

R[ 24]={1, 4, 9,6}

R[ 25]={1, 5, 4,1}

R[ 26]={1, 5, 4,4}

R[ 27]={1, 5, 6,1}

R[ 28]={1, 5, 6,5}

R[ 29]={1, 5,10,4}

R[ 30]={1, 5,10,5}

R[ 31]={1, 6, 5,2}

R[ 32]={1, 6, 5,4}

R[ 33]={1, 6, 7,2}

R[ 34]={1, 6, 7,6}

R[ 35]={1, 6,11,4}

R[ 36]={1, 6,11,6}

R[ 37]={1, 7, 8,3}

R[ 38]={1, 7, 8,5}

R[ 39]={1, 7, 9,3}

R[ 40]={1, 7, 9,6}

R[ 41]={1, 7,12,5}

R[ 42]={1, 7,12,6}

R[ 43]={1, 8,10,4}

R[ 44]={1, 8,10,5}

R[ 45]={1, 8,11,4}

R[ 46]={1, 8,11,6}

R[ 47]={1, 8,12,5}

R[ 48]={1, 8,12,6}

R[ 49]={2, 1, 1,1}

R[ 50]={2, 1, 1,2}

R[ 51]={2, 1, 2,1}

R[ 52]={2, 1, 2,3}

R[ 53]={2, 1, 3,2}

R[ 54]={2, 1, 3,3}

R[ 55]={2, 9, 1,1}

R[ 56]={2, 9, 1,2}

R[ 57]={2, 9,13,1}

R[ 58]={2, 9,13,7}

R[ 59]={2, 9,14,2}

R[ 60]={2, 9,14,7}

R[ 61]={2,10, 2,1}

R[ 62]={2,10, 2,3}

R[ 63]={2,10,15,1}

R[ 64]={2,10,15,8}

R[ 65]={2,10,17,3}

R[ 66]={2,10,17,8}

R[ 67]={2,11, 3,2}

R[ 68]={2,11, 3,3}

R[ 69]={2,11,16,2}

R[ 70]={2,11,16,9}

R[ 71]={2,11,18,3}

R[ 72]={2,11,18,9}

R[ 73]={2,21,13,1}

R[ 74]={2,21,13,7}

R[ 75]={2,21,15,1}

R[ 76]={2,21,15,8}

R[ 77]={2,21,37,7}

R[ 78]={2,21,37,8}

R[ 79]={2,22,14,2}

R[ 80]={2,22,14,7}

R[ 81]={2,22,16,2}

R[ 82]={2,22,16,9}

R[ 83]={2,22,38,7}

R[ 84]={2,22,38,9}

R[ 85]={2,23,17,3}

R[ 86]={2,23,17,8}

R[ 87]={2,23,18,3}

R[ 88]={2,23,18,9}

R[ 89]={2,23,39,8}

R[ 90]={2,23,39,9}

R[ 91]={2,45,37,7}

R[ 92]={2,45,37,8}

R[ 93]={2,45,38,7}

R[ 94]={2,45,38,9}

R[ 95]={2,45,39,8}

R[ 96]={2,45,39,9}

R[ 97]={3, 2, 1,1}

R[ 98]={3, 2, 1,2}

R[ 99]={3, 2, 4,1}

R[ 100]={3, 2, 4,4}

R[ 101]={3, 2, 5,2}

R[ 102]={3, 2, 5,4}

R[ 103]={3, 9, 1,1}

R[ 104]={3, 9, 1,2}

R[ 105]={3, 9,13,1}

R[ 106]={3, 9,13,7}

R[ 107]={3, 9,14,2}

R[ 108]={3, 9,14,7}

R[ 109]={3,12, 4,1}

R[ 110]={3,12, 4,4}

R[ 111]={3,12,19,1}

R[ 112]={3,12,19,10}

R[ 113]={3,12,25,4}

R[ 114]={3,12,25,10}

R[ 115]={3,13, 5,2}

R[ 116]={3,13, 5,4}

R[ 117]={3,13,20,2}

R[ 118]={3,13,20,11}

R[ 119]={3,13,26,4}

R[ 120]={3,13,26,11}

R[ 121]={3,24,13,1}

R[ 122]={3,24,13,7}

R[ 123]={3,24,19,1}

R[ 124]={3,24,19,10}

R[ 125]={3,24,40,7}

R[ 126]={3,24,40,10}

R[ 127]={3,25,14,2}

R[ 128]={3,25,14,7}

R[ 129]={3,25,20,2}

R[ 130]={3,25,20,11}

R[ 131]={3,25,41,7}

R[ 132]={3,25,41,11}

R[ 133]={3,33,25,4}

R[ 134]={3,33,25,10}

R[ 135]={3,33,26,4}

R[ 136]={3,33,26,11}

R[ 137]={3,33,46,10}

R[ 138]={3,33,46,11}

R[ 139]={3,46,40,7}

R[ 140]={3,46,40,10}

R[ 141]={3,46,41,7}

R[ 142]={3,46,41,11}

R[ 143]={3,46,46,10}

R[ 144]={3,46,46,11}

R[ 145]={4, 3, 2,1}

R[ 146]={4, 3, 2,3}

R[ 147]={4, 3, 6,1}

R[ 148]={4, 3, 6,5}

R[ 149]={4, 3, 8,3}

R[ 150]={4, 3, 8,5}

R[ 151]={4,10, 2,1}

R[ 152]={4,10, 2,3}

R[ 153]={4,10,15,1}

R[ 154]={4,10,15,8}

R[ 155]={4,10,17,3}

R[ 156]={4,10,17,8}

R[ 157]={4,14, 6,1}

R[ 158]={4,14, 6,5}

R[ 159]={4,14,21,1}

R[ 160]={4,14,21,12}

R[ 161]={4,14,27,5}

R[ 162]={4,14,27,12}

R[ 163]={4,16, 8,3}

R[ 164]={4,16, 8,5}

R[ 165]={4,16,23,3}

R[ 166]={4,16,23,14}

R[ 167]={4,16,29,5}

R[ 168]={4,16,29,14}

R[ 169]={4,26,15,1}

R[ 170]={4,26,15,8}

R[ 171]={4,26,21,1}

R[ 172]={4,26,21,12}

R[ 173]={4,26,42,8}

R[ 174]={4,26,42,12}

R[ 175]={4,28,17,3}

R[ 176]={4,28,17,8}

R[ 177]={4,28,23,3}

R[ 178]={4,28,23,14}

R[ 179]={4,28,44,8}

R[ 180]={4,28,44,14}

R[ 181]={4,34,27,5}

R[ 182]={4,34,27,12}

R[ 183]={4,34,29,5}

R[ 184]={4,34,29,14}

R[ 185]={4,34,49,12}

R[ 186]={4,34,49,14}

R[ 187]={4,47,42,8}

R[ 188]={4,47,42,12}

R[ 189]={4,47,44,8}

R[ 190]={4,47,44,14}

R[ 191]={4,47,49,12}

R[ 192]={4,47,49,14}

R[ 193]={5, 4, 3,2}

R[ 194]={5, 4, 3,3}

R[ 195]={5, 4, 7,2}

R[ 196]={5, 4, 7,6}

R[ 197]={5, 4, 9,3}

R[ 198]={5, 4, 9,6}

R[ 199]={5,11, 3,2}

R[ 200]={5,11, 3,3}

R[ 201]={5,11,16,2}

R[ 202]={5,11,16,9}

R[ 203]={5,11,18,3}

R[ 204]={5,11,18,9}

R[ 205]={5,15, 7,2}

R[ 206]={5,15, 7,6}

R[ 207]={5,15,22,2}

R[ 208]={5,15,22,13}

R[ 209]={5,15,28,6}

R[ 210]={5,15,28,13}

R[ 211]={5,17, 9,3}

R[ 212]={5,17, 9,6}

R[ 213]={5,17,24,3}

R[ 214]={5,17,24,15}

R[ 215]={5,17,30,6}

R[ 216]={5,17,30,15}

R[ 217]={5,27,16,2}

R[ 218]={5,27,16,9}

R[ 219]={5,27,22,2}

R[ 220]={5,27,22,13}

R[ 221]={5,27,43,9}

R[ 222]={5,27,43,13}

R[ 223]={5,29,18,3}

R[ 224]={5,29,18,9}

R[ 225]={5,29,24,3}

R[ 226]={5,29,24,15}

R[ 227]={5,29,45,9}

R[ 228]={5,29,45,15}

R[ 229]={5,35,28,6}

R[ 230]={5,35,28,13}

R[ 231]={5,35,30,6}

R[ 232]={5,35,30,15}

R[ 233]={5,35,50,13}

R[ 234]={5,35,50,15}

R[ 235]={5,48,43,9}

R[ 236]={5,48,43,13}

R[ 237]={5,48,45,9}

R[ 238]={5,48,45,15}

R[ 239]={5,48,50,13}

R[ 240]={5,48,50,15}

R[ 241]={6, 5, 4,1}

R[ 242]={6, 5, 4,4}

R[ 243]={6, 5, 6,1}

R[ 244]={6, 5, 6,5}

R[ 245]={6, 5,10,4}

R[ 246]={6, 5,10,5}

R[ 247]={6,12, 4,1}

R[ 248]={6,12, 4,4}

R[ 249]={6,12,19,1}

R[ 250]={6,12,19,10}

R[ 251]={6,12,25,4}

R[ 252]={6,12,25,10}

R[ 253]={6,14, 6,1}

R[ 254]={6,14, 6,5}

R[ 255]={6,14,21,1}

R[ 256]={6,14,21,12}

R[ 257]={6,14,27,5}

R[ 258]={6,14,27,12}

R[ 259]={6,18,10,4}

R[ 260]={6,18,10,5}

R[ 261]={6,18,31,4}

R[ 262]={6,18,31,16}

R[ 263]={6,18,33,5}

R[ 264]={6,18,33,16}

R[ 265]={6,30,19,1}

R[ 266]={6,30,19,10}

R[ 267]={6,30,21,1}

R[ 268]={6,30,21,12}

R[ 269]={6,30,47,10}

R[ 270]={6,30,47,12}

R[ 271]={6,36,25,4}

R[ 272]={6,36,25,10}

R[ 273]={6,36,31,4}

R[ 274]={6,36,31,16}

R[ 275]={6,36,52,10}

R[ 276]={6,36,52,16}

R[ 277]={6,38,27,5}

R[ 278]={6,38,27,12}

R[ 279]={6,38,33,5}

R[ 280]={6,38,33,16}

R[ 281]={6,38,54,12}

R[ 282]={6,38,54,16}

R[ 283]={6,49,47,10}

R[ 284]={6,49,47,12}

R[ 285]={6,49,52,10}

R[ 286]={6,49,52,16}

R[ 287]={6,49,54,12}

R[ 288]={6,49,54,16}

R[ 289]={7, 6, 5,2}

R[ 290]={7, 6, 5,4}

R[ 291]={7, 6, 7,2}

R[ 292]={7, 6, 7,6}

R[ 293]={7, 6,11,4}

R[ 294]={7, 6,11,6}

R[ 295]={7,13, 5,2}

R[ 296]={7,13, 5,4}

R[ 297]={7,13,20,2}

R[ 298]={7,13,20,11}

R[ 299]={7,13,26,4}

R[ 300]={7,13,26,11}

R[ 301]={7,15, 7,2}

R[ 302]={7,15, 7,6}

R[ 303]={7,15,22,2}

R[ 304]={7,15,22,13}

R[ 305]={7,15,28,6}

R[ 306]={7,15,28,13}

R[ 307]={7,19,11,4}

R[ 308]={7,19,11,6}

R[ 309]={7,19,32,4}

R[ 310]={7,19,32,17}

R[ 311]={7,19,34,6}

R[ 312]={7,19,34,17}

R[ 313]={7,31,20,2}

R[ 314]={7,31,20,11}

R[ 315]={7,31,22,2}

R[ 316]={7,31,22,13}

R[ 317]={7,31,48,11}

R[ 318]={7,31,48,13}

R[ 319]={7,37,26,4}

R[ 320]={7,37,26,11}

R[ 321]={7,37,32,4}

R[ 322]={7,37,32,17}

R[ 323]={7,37,53,11}

R[ 324]={7,37,53,17}

R[ 325]={7,39,28,6}

R[ 326]={7,39,28,13}

R[ 327]={7,39,34,6}

R[ 328]={7,39,34,17}

R[ 329]={7,39,55,13}

R[ 330]={7,39,55,17}

R[ 331]={7,50,48,11}

R[ 332]={7,50,48,13}

R[ 333]={7,50,53,11}

R[ 334]={7,50,53,17}

R[ 335]={7,50,55,13}

R[ 336]={7,50,55,17}

R[ 337]={8, 7, 8,3}

R[ 338]={8, 7, 8,5}

R[ 339]={8, 7, 9,3}

R[ 340]={8, 7, 9,6}

R[ 341]={8, 7,12,5}

R[ 342]={8, 7,12,6}

R[ 343]={8,16, 8,3}

R[ 344]={8,16, 8,5}

R[ 345]={8,16,23,3}

R[ 346]={8,16,23,14}

R[ 347]={8,16,29,5}

R[ 348]={8,16,29,14}

R[ 349]={8,17, 9,3}

R[ 350]={8,17, 9,6}

R[ 351]={8,17,24,3}

R[ 352]={8,17,24,15}

R[ 353]={8,17,30,6}

R[ 354]={8,17,30,15}

R[ 355]={8,20,12,5}

R[ 356]={8,20,12,6}

R[ 357]={8,20,35,5}

R[ 358]={8,20,35,18}

R[ 359]={8,20,36,6}

R[ 360]={8,20,36,18}

R[ 361]={8,32,23,3}

R[ 362]={8,32,23,14}

R[ 363]={8,32,24,3}

R[ 364]={8,32,24,15}

R[ 365]={8,32,51,14}

R[ 366]={8,32,51,15}

R[ 367]={8,40,29,5}

R[ 368]={8,40,29,14}

R[ 369]={8,40,35,5}

R[ 370]={8,40,35,18}

R[ 371]={8,40,56,14}

R[ 372]={8,40,56,18}

R[ 373]={8,41,30,6}

R[ 374]={8,41,30,15}

R[ 375]={8,41,36,6}

R[ 376]={8,41,36,18}

R[ 377]={8,41,57,15}

R[ 378]={8,41,57,18}

R[ 379]={8,51,51,14}

R[ 380]={8,51,51,15}

R[ 381]={8,51,56,14}

R[ 382]={8,51,56,18}

R[ 383]={8,51,57,15}

R[ 384]={8,51,57,18}

R[ 385]={9, 8,10,4}

R[ 386]={9, 8,10,5}

R[ 387]={9, 8,11,4}

R[ 388]={9, 8,11,6}

R[ 389]={9, 8,12,5}

R[ 390]={9, 8,12,6}

R[ 391]={9,18,10,4}

R[ 392]={9,18,10,5}

R[ 393]={9,18,31,4}

R[ 394]={9,18,31,16}

R[ 395]={9,18,33,5}

R[ 396]={9,18,33,16}

R[ 397]={9,19,11,4}

R[ 398]={9,19,11,6}

R[ 399]={9,19,32,4}

R[ 400]={9,19,32,17}

R[ 401]={9,19,34,6}

R[ 402]={9,19,34,17}

R[ 403]={9,20,12,5}

R[ 404]={9,20,12,6}

R[ 405]={9,20,35,5}

R[ 406]={9,20,35,18}

R[ 407]={9,20,36,6}

R[ 408]={9,20,36,18}

R[ 409]={9,42,31,4}

R[ 410]={9,42,31,16}

R[ 411]={9,42,32,4}

R[ 412]={9,42,32,17}

R[ 413]={9,42,58,16}

R[ 414]={9,42,58,17}

R[ 415]={9,43,33,5}

R[ 416]={9,43,33,16}

R[ 417]={9,43,35,5}

R[ 418]={9,43,35,18}

R[ 419]={9,43,59,16}

R[ 420]={9,43,59,18}

R[ 421]={9,44,34,6}

R[ 422]={9,44,34,17}

R[ 423]={9,44,36,6}

R[ 424]={9,44,36,18}

R[ 425]={9,44,60,17}

R[ 426]={9,44,60,18}

R[ 427]={9,52,58,16}

R[ 428]={9,52,58,17}

R[ 429]={9,52,59,16}

R[ 430]={9,52,59,18}

R[ 431]={9,52,60,17}

R[ 432]={9,52,60,18}

R[ 433]={10,21,13,1}

R[ 434]={10,21,13,7}

R[ 435]={10,21,15,1}

R[ 436]={10,21,15,8}

R[ 437]={10,21,37,7}

R[ 438]={10,21,37,8}

R[ 439]={10,24,13,1}

R[ 440]={10,24,13,7}

R[ 441]={10,24,19,1}

R[ 442]={10,24,19,10}

R[ 443]={10,24,40,7}

R[ 444]={10,24,40,10}

R[ 445]={10,26,15,1}

R[ 446]={10,26,15,8}

R[ 447]={10,26,21,1}

R[ 448]={10,26,21,12}

R[ 449]={10,26,42,8}

R[ 450]={10,26,42,12}

R[ 451]={10,30,19,1}

R[ 452]={10,30,19,10}

R[ 453]={10,30,21,1}

R[ 454]={10,30,21,12}

R[ 455]={10,30,47,10}

R[ 456]={10,30,47,12}

R[ 457]={10,53,37,7}

R[ 458]={10,53,37,8}

R[ 459]={10,53,61,7}

R[ 460]={10,53,61,19}

R[ 461]={10,53,63,8}

R[ 462]={10,53,63,19}

R[ 463]={10,56,40,7}

R[ 464]={10,56,40,10}

R[ 465]={10,56,61,7}

R[ 466]={10,56,61,19}

R[ 467]={10,56,67,10}

R[ 468]={10,56,67,19}

R[ 469]={10,58,42,8}

R[ 470]={10,58,42,12}

R[ 471]={10,58,63,8}

R[ 472]={10,58,63,19}

R[ 473]={10,58,69,12}

R[ 474]={10,58,69,19}

R[ 475]={10,62,47,10}

R[ 476]={10,62,47,12}

R[ 477]={10,62,67,10}

R[ 478]={10,62,67,19}

R[ 479]={10,62,69,12}

R[ 480]={10,62,69,19}

R[ 481]={11,22,14,2}

R[ 482]={11,22,14,7}

R[ 483]={11,22,16,2}

R[ 484]={11,22,16,9}

R[ 485]={11,22,38,7}

R[ 486]={11,22,38,9}

R[ 487]={11,25,14,2}

R[ 488]={11,25,14,7}

R[ 489]={11,25,20,2}

R[ 490]={11,25,20,11}

R[ 491]={11,25,41,7}

R[ 492]={11,25,41,11}

R[ 493]={11,27,16,2}

R[ 494]={11,27,16,9}

R[ 495]={11,27,22,2}

R[ 496]={11,27,22,13}

R[ 497]={11,27,43,9}

R[ 498]={11,27,43,13}

R[ 499]={11,31,20,2}

R[ 500]={11,31,20,11}

R[ 501]={11,31,22,2}

R[ 502]={11,31,22,13}

R[ 503]={11,31,48,11}

R[ 504]={11,31,48,13}

R[ 505]={11,54,38,7}

R[ 506]={11,54,38,9}

R[ 507]={11,54,62,7}

R[ 508]={11,54,62,20}

R[ 509]={11,54,64,9}

R[ 510]={11,54,64,20}

R[ 511]={11,57,41,7}

R[ 512]={11,57,41,11}

R[ 513]={11,57,62,7}

R[ 514]={11,57,62,20}

R[ 515]={11,57,68,11}

R[ 516]={11,57,68,20}

R[ 517]={11,59,43,9}

R[ 518]={11,59,43,13}

R[ 519]={11,59,64,9}

R[ 520]={11,59,64,20}

R[ 521]={11,59,70,13}

R[ 522]={11,59,70,20}

R[ 523]={11,63,48,11}

R[ 524]={11,63,48,13}

R[ 525]={11,63,68,11}

R[ 526]={11,63,68,20}

R[ 527]={11,63,70,13}

R[ 528]={11,63,70,20}

R[ 529]={12,23,17,3}

R[ 530]={12,23,17,8}

R[ 531]={12,23,18,3}

R[ 532]={12,23,18,9}

R[ 533]={12,23,39,8}

R[ 534]={12,23,39,9}

R[ 535]={12,28,17,3}

R[ 536]={12,28,17,8}

R[ 537]={12,28,23,3}

R[ 538]={12,28,23,14}

R[ 539]={12,28,44,8}

R[ 540]={12,28,44,14}

R[ 541]={12,29,18,3}

R[ 542]={12,29,18,9}

R[ 543]={12,29,24,3}

R[ 544]={12,29,24,15}

R[ 545]={12,29,45,9}

R[ 546]={12,29,45,15}

R[ 547]={12,32,23,3}

R[ 548]={12,32,23,14}

R[ 549]={12,32,24,3}

R[ 550]={12,32,24,15}

R[ 551]={12,32,51,14}

R[ 552]={12,32,51,15}

R[ 553]={12,55,39,8}

R[ 554]={12,55,39,9}

R[ 555]={12,55,65,8}

R[ 556]={12,55,65,21}

R[ 557]={12,55,66,9}

R[ 558]={12,55,66,21}

R[ 559]={12,60,44,8}

R[ 560]={12,60,44,14}

R[ 561]={12,60,65,8}

R[ 562]={12,60,65,21}

R[ 563]={12,60,71,14}

R[ 564]={12,60,71,21}

R[ 565]={12,61,45,9}

R[ 566]={12,61,45,15}

R[ 567]={12,61,66,9}

R[ 568]={12,61,66,21}

R[ 569]={12,61,72,15}

R[ 570]={12,61,72,21}

R[ 571]={12,64,51,14}

R[ 572]={12,64,51,15}

R[ 573]={12,64,71,14}

R[ 574]={12,64,71,21}

R[ 575]={12,64,72,15}

R[ 576]={12,64,72,21}

R[ 577]={13,33,25,4}

R[ 578]={13,33,25,10}

R[ 579]={13,33,26,4}

R[ 580]={13,33,26,11}

R[ 581]={13,33,46,10}

R[ 582]={13,33,46,11}

R[ 583]={13,36,25,4}

R[ 584]={13,36,25,10}

R[ 585]={13,36,31,4}

R[ 586]={13,36,31,16}

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R[1100]={23,96,94,23}

R[1101]={23,96,95,22}

R[1102]={23,96,95,24}

R[1103]={23,96,96,23}

R[1104]={23,96,96,24}

R[1105]={24,89,85,19}

R[1106]={24,89,85,20}

R[1107]={24,89,86,19}

R[1108]={24,89,86,21}

R[1109]={24,89,87,20}

R[1110]={24,89,87,21}

R[1111]={24,90,85,19}

R[1112]={24,90,85,20}

R[1113]={24,90,88,19}

R[1114]={24,90,88,22}

R[1115]={24,90,89,20}

R[1116]={24,90,89,22}

R[1117]={24,91,86,19}

R[1118]={24,91,86,21}

R[1119]={24,91,90,19}

R[1120]={24,91,90,23}

R[1121]={24,91,92,21}

R[1122]={24,91,92,23}

R[1123]={24,92,87,20}

R[1124]={24,92,87,21}

R[1125]={24,92,91,20}

R[1126]={24,92,91,24}

R[1127]={24,92,93,21}

R[1128]={24,92,93,24}

R[1129]={24,93,88,19}

R[1130]={24,93,88,22}

R[1131]={24,93,90,19}

R[1132]={24,93,90,23}

R[1133]={24,93,94,22}

R[1134]={24,93,94,23}

R[1135]={24,94,89,20}

R[1136]={24,94,89,22}

R[1137]={24,94,91,20}

R[1138]={24,94,91,24}

R[1139]={24,94,95,22}

R[1140]={24,94,95,24}

R[1141]={24,95,92,21}

R[1142]={24,95,92,23}

R[1143]={24,95,93,21}

R[1144]={24,95,93,24}

R[1145]={24,95,96,23}

R[1146]={24,95,96,24}

R[1147]={24,96,94,22}

R[1148]={24,96,94,23}

R[1149]={24,96,95,22}

R[1150]={24,96,95,24}

R[1151]={24,96,96,23}

R[1152]={24,96,96,24}