2022.04.01 MATH 3600 (3n+1)/2 levels

92 days ago by calkin

How fast do the levels of the tree for $g(n)$ grow?

def gparent(n): # this will return a list of the values m for which g(m)=n if mod(n,3)==2: return([Integer((2*n-1)/3),Integer(2*n)]) else: return([Integer(2*n)]) 
       
gparent(3) 
       
[6]
[6]
gparent(5) 
       
[3, 10]
[3, 10]
gparent(32) 
       
[21, 64]
[21, 64]

To compute a list of the levels, start by populating it with the first three levels.

Now, each time we look at the top level (i.e. the last level added)  and for each 

element of the level, compute its gparents, and add them to the next level.  At the end,

append the next level to the list glevels.

glevels=[[1],[2],[4]] glevelsizes=[[0,1],[1,1],[2,1]] i=3 while i<50: lastlevel=glevels[-1] nextlevel=[] for j in lastlevel: jparents=gparent(j) for k in jparents: nextlevel.append(k) glevels.append(nextlevel) glevelsizes.append([i,Integer(len(nextlevel))]) i+=1 print(glevelsizes) 
       
[[0, 1], [1, 1], [2, 1], [3, 1], [4, 2], [5, 3], [6, 4], [7, 5], [8, 6],
[9, 8], [10, 12], [11, 18], [12, 24], [13, 31], [14, 39], [15, 50], [16,
68], [17, 91], [18, 120], [19, 159], [20, 211], [21, 282], [22, 381],
[23, 505], [24, 665], [25, 885], [26, 1187], [27, 1590], [28, 2122],
[29, 2829], [30, 3765], [31, 5014], [32, 6682], [33, 8902], [34, 11878],
[35, 15844], [36, 21122], [37, 28150], [38, 37536], [39, 50067], [40,
66763], [41, 89009], [42, 118631], [43, 158171], [44, 210939], [45,
281334], [46, 375129], [47, 500106], [48, 666725], [49, 888947]]
[[0, 1], [1, 1], [2, 1], [3, 1], [4, 2], [5, 3], [6, 4], [7, 5], [8, 6], [9, 8], [10, 12], [11, 18], [12, 24], [13, 31], [14, 39], [15, 50], [16, 68], [17, 91], [18, 120], [19, 159], [20, 211], [21, 282], [22, 381], [23, 505], [24, 665], [25, 885], [26, 1187], [27, 1590], [28, 2122], [29, 2829], [30, 3765], [31, 5014], [32, 6682], [33, 8902], [34, 11878], [35, 15844], [36, 21122], [37, 28150], [38, 37536], [39, 50067], [40, 66763], [41, 89009], [42, 118631], [43, 158171], [44, 210939], [45, 281334], [46, 375129], [47, 500106], [48, 666725], [49, 888947]]
list_plot(glevelsizes) 
       
for i in srange(5,len(glevelsizes)-1): #print(glevelsizes[i+1][1],glevelsizes[i][1]) r=glevelsizes[i+1][1]/glevelsizes[i][1] print(r.n()) 
       
1.33333333333333
1.25000000000000
1.20000000000000
1.33333333333333
1.50000000000000
1.50000000000000
1.33333333333333
1.29166666666667
1.25806451612903
1.28205128205128
1.36000000000000
1.33823529411765
1.31868131868132
1.32500000000000
1.32704402515723
1.33649289099526
1.35106382978723
1.32545931758530
1.31683168316832
1.33082706766917
1.34124293785311
1.33951137320977
1.33459119496855
1.33317624882187
1.33085896076352
1.33173970783533
1.33266852812126
1.33223585752769
1.33430689732644
1.33389459504967
1.33312294875032
1.33273364264748
1.33342806394316
1.33383951406650
1.33347314598438
1.33320851369771
1.33279780696334
1.33330242516711
1.33361362070165
1.33372207130972
1.33339375973043
1.33315739385651
1.33316736851787
1.33330383591436
1.33333333333333
1.25000000000000
1.20000000000000
1.33333333333333
1.50000000000000
1.50000000000000
1.33333333333333
1.29166666666667
1.25806451612903
1.28205128205128
1.36000000000000
1.33823529411765
1.31868131868132
1.32500000000000
1.32704402515723
1.33649289099526
1.35106382978723
1.32545931758530
1.31683168316832
1.33082706766917
1.34124293785311
1.33951137320977
1.33459119496855
1.33317624882187
1.33085896076352
1.33173970783533
1.33266852812126
1.33223585752769
1.33430689732644
1.33389459504967
1.33312294875032
1.33273364264748
1.33342806394316
1.33383951406650
1.33347314598438
1.33320851369771
1.33279780696334
1.33330242516711
1.33361362070165
1.33372207130972
1.33339375973043
1.33315739385651
1.33316736851787
1.33330383591436
1.34^10 
       
18.6658591186100
18.6658591186100
for i in srange(5,len(glevelsizes)): #print(glevelsizes[i+1][1],glevelsizes[i][1]) r=2*glevelsizes[i][1]/(4/3)^(i+1) print(r.n()) 
       
1.06787109375000
1.06787109375000
1.00112915039062
0.901016235351562
0.901016235351562
1.01364326477051
1.14034867286682
1.14034867286682
1.10471277683973
1.04234995879233
1.00225957576185
1.02230476727709
1.02606324068620
1.01478782045888
1.00844539658101
1.00368857867261
1.00606698762681
1.01944553799419
1.01342519032887
1.00088279935945
0.999001440714038
1.00492772044709
1.00958408309448
1.01053652090872
1.01041746618194
1.00854235423516
1.00733692512651
1.00683466299791
1.00600593048571
1.00674048884879
1.00716427251980
1.00700535364317
1.00655243484478
1.00662394833927
1.00700609857541
1.00711169271961
1.00701741223372
1.00661294894925
1.00658961452899
1.00680121529451
1.00709475169480
1.00714039302533
1.00700749620996
1.00688215035000
1.00685987503151
1.06787109375000
1.06787109375000
1.00112915039062
0.901016235351562
0.901016235351562
1.01364326477051
1.14034867286682
1.14034867286682
1.10471277683973
1.04234995879233
1.00225957576185
1.02230476727709
1.02606324068620
1.01478782045888
1.00844539658101
1.00368857867261
1.00606698762681
1.01944553799419
1.01342519032887
1.00088279935945
0.999001440714038
1.00492772044709
1.00958408309448
1.01053652090872
1.01041746618194
1.00854235423516
1.00733692512651
1.00683466299791
1.00600593048571
1.00674048884879
1.00716427251980
1.00700535364317
1.00655243484478
1.00662394833927
1.00700609857541
1.00711169271961
1.00701741223372
1.00661294894925
1.00658961452899
1.00680121529451
1.00709475169480
1.00714039302533
1.00700749620996
1.00688215035000
1.00685987503151