In[]:=
NestGraph[x|->If[x<2,{1},{x^2-1,x-2}],2,8]
Out[]=
In[]:=
data=
;
In[]:=
ListLinePlot[Differences[Drop[data,-400],2],PlotRange->All]
Out[]=
100
200
300
400
500
600
-3e+6
-2e+6
-1e+6
1e+6
2e+6
3e+6
4e+6
In[]:=
ListLogPlot[Differences[Drop[data,-400],2],PlotRange->All,Joined->True]
Out[]=
In[]:=
VertexList[NestGraph[n{2n+1,3n+1},0,15]];
In[]:=
Length[%]
Out[]=
29806
In[]:=
Take[Sort@VertexList[NestGraph[n{2n+1,3n+1},0,15]],25]
Out[]=
{0,1,3,4,7,9,10,13,15,19,21,22,27,28,31,39,40,43,45,46,55,57,58,63,64}
In[]:=
Take[Sort@VertexList[NestGraph[n{2n+1,3n+1},0,16]],500]
Out[]=
{0,1,3,4,7,9,10,13,15,19,21,22,27,28,31,39,40,43,45,46,55,57,58,63,64,67,79,81,82,85,87,91,93,94,111,115,117,118,121,127,129,130,135,136,139,159,163,165,166,171,172,175,183,187,189,190,193,202,223,231,235,237,238,243,244,247,255,256,259,261,262,271,273,274,279,280,283,319,327,331,333,334,343,345,346,351,352,355,364,367,375,379,381,382,387,388,391,405,406,409,418,447,463,471,475,477,478,487,489,490,495,496,499,511,513,514,517,519,523,525,526,543,547,549,550,559,561,562,567,568,571,580,607,639,655,663,667,669,670,687,691,693,694,703,705,706,711,712,715,729,730,733,735,742,751,759,763,765,766,769,775,777,778,783,784,787,811,813,814,819,820,823,837,838,841,850,895,927,943,951,955,957,958,975,979,981,982,991,993,994,999,1000,1003,1023,1027,1029,1030,1035,1036,1039,1047,1051,1053,1054,1057,1066,1087,1093,1095,1099,1101,1102,1119,1123,1125,1126,1135,1137,1138,1143,1144,1147,1161,1162,1165,1174,1215,1216,1219,1228,1255,1279,1311,1327,1335,1339,1341,1342,1375,1383,1387,1389,1390,1407,1411,1413,1414,1423,1425,1426,1431,1432,1435,1459,1461,1462,1467,1468,1471,1485,1486,1489,1498,1503,1519,1527,1531,1533,1534,1539,1540,1543,1551,1552,1555,1557,1558,1567,1569,1570,1575,1576,1579,1623,1627,1629,1630,1639,1641,1642,1647,1648,1651,1675,1677,1678,1683,1684,1687,1701,1702,1705,1714,1741,1791,1822,1855,1887,1903,1911,1915,1917,1918,1951,1959,1963,1965,1966,1983,1987,1989,1990,1999,2001,2002,2007,2008,2011,2047,2055,2059,2061,2062,2071,2073,2074,2079,2080,2083,2095,2103,2107,2109,2110,2115,2116,2119,2133,2134,2137,2146,2175,2187,2188,2191,2199,2200,2203,2205,2206,2227,2239,2247,2251,2253,2254,2271,2275,2277,2278,2287,2289,2290,2295,2296,2299,2308,2323,2325,2326,2331,2332,2335,2349,2350,2353,2362,2431,2433,2434,2439,2440,2443,2457,2458,2461,2470,2511,2512,2515,2524,2551,2559,2623,2655,2671,2679,2683,2685,2686,2751,2767,2775,2779,2781,2782,2815,2823,2827,2829,2830,2847,2851,2853,2854,2863,2865,2866,2871,2872,2875,2919,2923,2925,2926,2935,2937,2938,2943,2944,2947,2971,2973,2974,2979,2980,2983,2997,2998,3001,3007,3010,3039,3055,3063,3067,3069,3070,3079,3081,3082,3087,3088,3091,3103,3105,3106,3109,3111,3115,3117,3118,3135,3139,3141,3142,3151,3153,3154,3159,3160,3163,3172,3199,3247,3255,3259,3261,3262,3279,3280,3283,3285,3286,3295,3297,3298,3303,3304,3307}
In[]:=
Take[Sort@VertexList[NestGraph[n{2n+1,3n+1},0,16]],1000]==Take[Sort@VertexList[NestGraph[n{2n+1,3n+1},0,15]],1000]
Out[]=
True
In[]:=
Min[Complement[Sort@VertexList[NestGraph[n{2n+1,3n+1},0,16]],Sort@VertexList[NestGraph[n{2n+1,3n+1},0,15]]]]
Out[]=
96255
In[]:=
FirstPosition[Sort@VertexList[NestGraph[n{2n+1,3n+1},1,15]],x_/;x>96255]
Out[]=
{6728}
In[]:=
ListLinePlot[Range[5000]/Take[Sort@VertexList[NestGraph[n{2n+1,3n+1},1,15]],5000]]
Out[]=
1000
2000
3000
4000
5000
0.05
0.10
0.15
0.20
In[]:=
ListLinePlot[Range[5000]/Take[Sort@VertexList[NestGraph[n{2n+1,3n+1},1,15]],5000],Frame->True,AspectRatio1/3,Filling->Axis,FillingStyle->LightBlue,ScalingFunctions{"Log",Identity}]
Out[]=
In[]:=
ListLinePlot[Range[5000]/Take[Sort@VertexList[NestGraph[n{2n+1,3n+1},1,15]],5000],Frame->True,AspectRatio1/3,Filling->Axis,FillingStyle->LightBlue,ScalingFunctions{"Log","Log"}]
Out[]=
In[]:=
Histogram[Differences[Take[Sort@VertexList[NestGraph[n{2n+1,3n+1},1,15]],5000]],{1},PlotRange->All]
Out[]=
0
200
400
600
800
1000
0
200
400
600
800
1000
1200
It’s perfectly possible to get more complex patterns, though. For example,
gives
Ed: can you find the form here?
In the general case
several types of patterns are seen:

Confluence Investigation

Variable Numbers of Outputs

Branchial Graphs

The branchial graph is trivial because of the foliation....

Floor functions

Negative Numbers

Merging

Formatting

Confluence testing

Might want to find some overlap ... not necessarily the first overlap...

A layer can be dropped if the min of a new layer is larger than the max of the previous layer....

Ed’s Observation

What is the final approach to a merger?

Possible paths

The set of possible sequences of transformations is given by a regular language [i.e. paths in a finite graph]

What is the regex for this?

Infinite collection of possible sequence pairs ; do any of them correspond?

Similar to Post Correspondence Problem
Do all possible branch pairs converge may be undecidable [[[ what type of sentence ? ]]]
for all branch pairs there exists a value t such that XXXX

Needs universality: you can embed any computation in the pair of initial conditions

Nik’s pruner

Branchial Graphs

Number Theory

“Remainder graphs”
Base + modulus....
Generating remainder by going left to right in a decimal number:
https://demonstrations.wolfram.com/RemainderGraphs/
Each digit corresponds to the number of blue arrows; then you jump by a red arrow....

Partitions

Rulial case.....

Others

each “efflorescence” corresponds to a prime gap