In[]:=
d0=Differences[{6,6,7,7,8,9,8,9,10,11,12,13,13,14,16,17,17,20,21,22,22,23,27,28,28,31,32,33,33,34,35,37,38,38,41,42,43,43,44,46,47,49,50,50,53,54,55,55,56,60,62,63,63,66,67,68,68,69,70,71,73,74,74,77,78,79,79,80,84,85,87,88,88,91,92,93,93,94,96,97,98,100,101,101,104,105,106,106,107,112,113,114,116,117,117,120,121,122,122,123,125,127,128,129,131,132,132,135,136,137,137,138,143,145,146,147,149,150,150,153,154,155,155,156,158,160,162,163,164,166,167,167,170,171,172,172,173,178,180,182,183,184,186,187,187,190,191,192,192,193,195}]
Out[]=
{0,1,0,1,1,-1,1,1,1,1,1,0,1,2,1,0,3,1,1,0,1,4,1,0,3,1,1,0,1,1,2,1,0,3,1,1,0,1,2,1,2,1,0,3,1,1,0,1,4,2,1,0,3,1,1,0,1,1,1,2,1,0,3,1,1,0,1,4,1,2,1,0,3,1,1,0,1,2,1,1,2,1,0,3,1,1,0,1,5,1,1,2,1,0,3,1,1,0,1,2,2,1,1,2,1,0,3,1,1,0,1,5,2,1,1,2,1,0,3,1,1,0,1,2,2,2,1,1,2,1,0,3,1,1,0,1,5,2,2,1,1,2,1,0,3,1,1,0,1,2}
In[]:=
Position[d0,0]//Flatten
Out[]=
{1,3,12,16,20,24,28,33,37,43,47,52,56,62,66,72,76,83,87,94,98,106,110,118,122,131,135,144,148}
In[]:=
Differences[%]
Out[]=
{2,9,4,4,4,4,5,4,6,4,5,4,6,4,6,4,7,4,7,4,8,4,8,4,9,4,9,4}
In[]:=
sd[a_,b_]:={-Length[Complement[a,b]],Length[Complement[b,a]]}
In[]:=
sd@@@Partition[Union[Level[#,{0,Infinity},HeadsTrue]]&/@CombinatorEvolveList[s[s][s][s[s]][s][s],100],2,1]
Out[]=
{{-4,4},{-3,4},{-3,3},{-2,3},{-2,3},{-3,2},{-2,3},{-2,3},{-2,3},{-3,4},{-4,5},{-4,4},{-3,4},{-3,5},{-4,5},{-4,4},{-3,6},{-5,6},{-6,7},{-6,6},{-5,6},{-5,9},{-8,9},{-8,8},{-7,10},{-9,10},{-10,11},{-10,10},{-9,10},{-9,10},{-9,11},{-10,11},{-10,10},{-9,12},{-11,12},{-12,13},{-12,12},{-11,12},{-11,13},{-12,13},{-12,14},{-13,14},{-13,13},{-12,15},{-14,15},{-15,16},{-15,15},{-14,15},{-14,18},{-17,19},{-18,19},{-18,18},{-17,20},{-19,20},{-20,21},{-20,20},{-19,20},{-19,20},{-19,20},{-19,21},{-20,21},{-20,20},{-19,22},{-21,22},{-22,23},{-22,22},{-21,22},{-21,25},{-24,25},{-24,26},{-25,26},{-25,25},{-24,27},{-26,27},{-27,28},{-27,27},{-26,27},{-26,28},{-27,28},{-27,28},{-27,29},{-28,29},{-28,28},{-27,30},{-29,30},{-30,31},{-30,30},{-29,30},{-29,34},{-33,34},{-33,34},{-33,35},{-34,35},{-34,34},{-33,36},{-35,36},{-36,37},{-36,36},{-35,36},{-35,37}}
In[]:=
ListStepPlot[Transpose[%]]
Out[]=
20
40
60
80
100
-40
-20
20
40
In[]:=
Plus@@@%94
Out[]=
{0,1,0,1,1,-1,1,1,1,1,1,0,1,2,1,0,3,1,1,0,1,4,1,0,3,1,1,0,1,1,2,1,0,3,1,1,0,1,2,1,2,1,0,3,1,1,0,1,4,2,1,0,3,1,1,0,1,1,1,2,1,0,3,1,1,0,1,4,1,2,1,0,3,1,1,0,1,2,1,1,2,1,0,3,1,1,0,1,5,1,1,2,1,0,3,1,1,0,1,2}
In[]:=
Differences/@Transpose[%94]
Out[]=
{{1,0,1,0,-1,1,0,0,-1,-1,0,1,0,-1,0,1,-2,-1,0,1,0,-3,0,1,-2,-1,0,1,0,0,-1,0,1,-2,-1,0,1,0,-1,0,-1,0,1,-2,-1,0,1,0,-3,-1,0,1,-2,-1,0,1,0,0,0,-1,0,1,-2,-1,0,1,0,-3,0,-1,0,1,-2,-1,0,1,0,-1,0,0,-1,0,1,-2,-1,0,1,0,-4,0,0,-1,0,1,-2,-1,0,1,0},{0,-1,0,0,-1,1,0,0,1,1,-1,0,1,0,-1,2,0,1,-1,0,3,0,-1,2,0,1,-1,0,0,1,0,-1,2,0,1,-1,0,1,0,1,0,-1,2,0,1,-1,0,3,1,0,-1,2,0,1,-1,0,0,0,1,0,-1,2,0,1,-1,0,3,0,1,0,-1,2,0,1,-1,0,1,0,0,1,0,-1,2,0,1,-1,0,4,0,0,1,0,-1,2,0,1,-1,0,1}}
In[]:=
ListStepPlot[%]
Out[]=
20
40
60
80
100
-4
-2
2
4
In[]:=
ListStepPlot/@%97
Out[]=

20
40
60
80
100
-4
-3
-2
-1
1
,
20
40
60
80
100
-1
0
1
2
3
4

In[]:=
s7t300=sd@@@Partition[Union[Level[#,{0,Infinity},HeadsTrue]]&/@CombinatorEvolveList[s[s][s][s[s]][s][s],300],2,1]
Out[]=
In[]:=
ListStepPlot@*Differences/@Transpose[%]
Out[]=

50
100
150
200
250
300
-2.5
-2.0
-1.5
-1.0
-0.5
0.5
1.0
,
50
100
150
200
250
300
-1.0
-0.5
0.5
1.0
1.5
2.0
2.5


Ed’s Formula

Full Collection

S cases