The “ruler” case
The “ruler” case
In[]:=
ListStepPlot[Differences[SKCombinatorLeftmostOutermostLeafCounts[s[s[s]][s][s[s]][s][k],50000]],FrameTrue,JoinedTrue,FillingAxis,FillingStyleGrayLevel[.9],AspectRatio1/5,PlotRangeAll]
Out[]=
In[]:=
peakmax=First/@Split[FoldList[Max,Differences[SKCombinatorLeftmostOutermostLeafCounts[s[s[s]][s][s[s]][s][k],10000]]]]
Out[]=
{1,2,5,6,13,14,15,16,32,53,54,108,202,210,406,414,814,822,1630,1638,3262,3270,6526,6534,13054,13062}
In[]:=
FindSequenceFunction[%]
Out[]=
$Aborted
In[]:=
Differences[peakmax]
Out[]=
{1,3,1,7,1,1,1,16,21,1,54,94,8,196,8,400,8,808,8,1624,8,3256,8,6520,8}
In[]:=
peakmax=First/@Split[FoldList[Max,Differences[SKCombinatorLeftmostOutermostLeafCounts[s[s[s]][s][s[s]][s][k],100000]]]]
Out[]=
{1,2,5,6,13,14,15,16,32,53,54,108,202,210,406,414,814,822,1630,1638,3262,3270,6526,6534,13054,13062,26110,26118,52222,52230,104446,104454,208894}
In[]:=
Differences[peakmax]
Out[]=
{1,3,1,7,1,1,1,16,21,1,54,94,8,196,8,400,8,808,8,1624,8,3256,8,6520,8,13048,8,26104,8,52216,8,104440}
In[]:=
{8,196,8,400,8,808,8,1624,8,3256,8,6520,8,13048,8,26104,8,52216,8,104440}[[2;;;;2]]
Out[]=
{196,400,808,1624,3256,6520,13048,26104,52216,104440}
In[]:=
Ratios[%]//N
Out[]=
{2.04082,2.02,2.0099,2.00493,2.00246,2.00123,2.00061,2.00031,2.00015}
In[]:=
FindLinearRecurrence[{196,400,808,1624,3256,6520,13048,26104,52216,104440}]
Out[]=
{3,-2}
In[]:=
FindSequenceFunction[{196,400,808,1624,3256,6520,13048,26104,52216,104440},n]
Out[]=
2(-4+51)
n
2
Successive highest peaks have heights
2(-4+51)
n
2
In[]:=
bbs=Differences[SKCombinatorLeftmostOutermostLeafCounts[s[s[s]][s][s[s]][s][k],100000]];
In[]:=
Take[bbs,-10]
Out[]=
{33,34,0,34,68,-70,-109,0,0,0}
In[]:=
Position[bbs,#]&/@{196,400,808,1624,3256,6520,13048,26104,52216,104440}
Out[]=
{{},{},{},{},{},{},{},{},{},{}}
In[]:=
Histogram[bbs,Automatic,{"Log","Count"}]
Out[]=
In[]:=
Select[peakmax,#>202&]
Out[]=
{210,406,414,814,822,1630,1638,3262,3270,6526,6534,13054,13062,26110,26118,52222,52230,104446,104454,208894}
In[]:=
%[[;;;;2]]
Out[]=
{210,414,822,1638,3270,6534,13062,26118,52230,104454}
In[]:=
FindSequenceFunction[%,n]
Out[]=
6(1+17)
n
2
In[]:=
TraditionalForm[%187]
Out[]//TraditionalForm=
6(17+1)
n
2
In[]:=
Position[bbs,#]&/@%186
Out[]=
{{{110}},{{256}},{{546}},{{1124}},{{2278}},{{4584}},{{9194}},{{18412}},{{36846}},{{73712}}}
In[]:=
FindSequenceFunction[Flatten[%],n]
Out[]=
2(-18+9+n)
2+n
2
In[]:=
TraditionalForm[%]
Out[]//TraditionalForm=
2(n+9-18)
n+2
2
In[]:=
Sort[Counts[bbs]]
Out[]=
11,21,51,141,121,71,151,-341,531,1081,-1101,-691,1001,-2041,-191,-681,1041,2101,-2121,-1371,-4081,-1361,2061,4141,-4161,-2731,-8161,-2721,4101,8221,-8241,-5451,-16321,-5441,8181,16381,-16401,-10891,-32641,-10881,16341,32701,-32721,-21771,-65281,-21761,32661,65341,-65361,-43531,-130561,-43521,65301,130621,-130641,-87051,-261121,-87041,130581,261181,-261201,-174091,-522241,-174081,261141,522301,-522321,-348171,-1044481,696321,-696341,-348161,522261,1044541,-1044561,696311,-696331,2088941,-2088961,-522311,132,112,542,1012,2022,1052,2032,4062,2072,4072,8142,4112,8152,16302,8192,16312,32622,16352,32632,65262,32672,65272,130542,65312,130552,261102,130592,261112,522222,-348182,261152,348152,522232,1044462,348142,522272,1044472,63,323,663,1343,2703,5423,10863,21743,43503,87023,174063,348164,-261194,-174105,174076,87039,1740810,-1306312,-870614,435117,870424,-653529,-435431,217534,435257,-327161,-217863,108766,2176123,-1639126,-1090128,543131,1088252,-823256,-546258,271261,544512,-415517,-274519,135522,2721033,-2111038,-1381040,671043,1362076,-1092080,-702081,332083,684159,-584163,-364164,164166,348324,178327,88329,-158329,033327
In[]:=
KeySort[Counts[bbs]]
Out[]=
-2088961,-1044561,-1044481,-696341,-696331,-522321,-522311,-522241,-348182,-348171,-348161,-261201,-261194,-261121,-174105,-174091,-174081,-130641,-1306312,-130561,-870614,-87051,-87041,-65361,-653529,-65281,-435431,-43531,-43521,-32721,-327161,-32641,-217863,-21771,-21761,-16401,-1639126,-16321,-1090128,-10891,-10881,-8241,-823256,-8161,-546258,-5451,-5441,-4161,-415517,-4081,-274519,-2731,-2721,-2121,-2111038,-2041,-1381040,-1371,-1361,-1101,-1092080,-702081,-691,-681,-584163,-364164,-341,-191,-158329,033327,11,21,51,63,71,88329,112,121,132,141,151,164166,178327,323,332083,348324,531,542,663,671043,684159,1001,1012,1041,1052,1081,1343,135522,1362076,2022,2032,2061,2072,2101,2703,271261,2721033,4062,4072,4101,4112,4141,5423,543131,544512,8142,8152,8181,8192,8221,10863,108766,1088252,16302,16312,16341,16352,16381,21743,217534,2176123,32622,32632,32661,32672,32701,43503,435117,435257,65262,65272,65301,65312,65341,87023,87039,870424,130542,130552,130581,130592,130621,174063,174076,1740810,261102,261112,261141,261152,261181,348142,348152,348164,522222,522232,522261,522272,522301,696311,696321,1044462,1044472,1044541,2088941
In[]:=
Histogram[Select[bbs,-100<#<100&],{1}]
Out[]=
Comb
Comb
Comb 2
Comb 2
More
More
Spiked
Spiked
Earlier case:
Earlier case: