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[]=
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[]=
In[]:=
ListLinePlot[Range[5000]/Take[Sort@VertexList[NestGraph[n{2n+1,3n+1},1,15]],5000],Frame->True,AspectRatio1/3,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,AspectRatio1/3,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[]=
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
Confluence Investigation
Variable Numbers of Outputs
Variable Numbers of Outputs
Branchial Graphs
Branchial Graphs
The branchial graph is trivial because of the foliation....
Floor functions
Floor functions
Negative Numbers
Negative Numbers
Merging
Merging
Formatting
Formatting
Confluence testing
Confluence testing
Might want to find some overlap ... not necessarily the first overlap...
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
Ed’s Observation
What is the final approach to a merger?
What is the final approach to a merger?
Possible paths
Possible paths
The set of possible sequences of transformations is given by a regular language [i.e. paths in a finite graph]
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?
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
Needs universality: you can embed any computation in the pair of initial conditions
Nik’s pruner
Nik’s pruner
Branchial Graphs
Branchial Graphs
Number Theory
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
Partitions
Rulial case.....
Others
Others
each “efflorescence” corresponds to a prime gap