mot[n_?OddQ]=0;mot[n_]:=1+mot[n/2]
init={u[1,0],u[1,1]};
step[l_]:=Union[Flatten[{l,l/.{u[k_,0]:>{u[k+1,0],u[k,1]},u[k_,h_]:>With[{p=mot[k]},If[h==p+1,{u[k+2^p,h],If[h==1,u[k+1,0],d[k,h-1]]},{u[k,h+1],d[k,h]}]],d[k_,h_]:>{u[k+2^(h-1),h],If[h==1,u[k+1,0],d[k,h-1]]}}}]]
Differences[l_]:=Drop[l,1]-Drop[l,-1]
step[init]
{u[1,1],u[2,0],u[2,1]}
Nest[step,init,3]
{d[2,1],u[1,0],u[1,1],u[2,0],u[2,1],u[2,2],u[3,0],u[3,1],u[4,0],u[4,2]}
Length/@NestList[step,init,5]
{2,4,7,10,14,19}
Differences[%]
{2,3,3,4,5}
sizesTo[n_]:=Differences[Length/@NestList[step,init,n]]
sizesTo[10]
{2,3,3,4,5,6,5,8,9,13}
sizesTo[20]//ListPlot
⁃Graphics⁃
ratios[l_]:=Drop[l,1]/Drop[l,-1]
ratios[sizesTo[20]]//N
{1.5,1.,1.33333,1.25,1.2,0.833333,1.6,1.125,1.44444,1.15385,1.2,1.27778,1.30435,1.1,1.27273,1.2619,1.20755,1.17188,1.28}
ratios[sizesTo[25]]//N
{1.5,1.,1.33333,1.25,1.2,0.833333,1.6,1.125,1.44444,1.15385,1.2,1.27778,1.30435,1.1,1.27273,1.2619,1.20755,1.17188,1.28,1.21875,1.19658,1.22143,1.25146,1.20093}
ListPlot[%]
⁃Graphics⁃
ratios[sizesTo[30]]//N
{1.5,1.,1.33333,1.25,1.2,0.833333,1.6,1.125,1.44444,1.15385,1.2,1.27778,1.30435,1.1,1.27273,1.2619,1.20755,1.17188,1.28,1.21875,1.19658,1.22143,1.25146,1.20093,1.21401,1.23397,1.22597,1.20551,1.22671}
ratios[sizesTo[35]]//N
{1.5,1.,1.33333,1.25,1.2,0.833333,1.6,1.125,1.44444,1.15385,1.2,1.27778,1.30435,1.1,1.27273,1.2619,1.20755,1.17188,1.28,1.21875,1.19658,1.22143,1.25146,1.20093,1.21401,1.23397,1.22597,1.20551,1.22671,1.22779,1.21587,1.21593,1.2281,1.22044}
ListPlot[%]
⁃Graphics⁃
sizesTo[20]
{2,3,3,4,5,6,5,8,9,13,15,18,23,30,33,42,53,64,75,96}
Partition[%67,8,1]
{{2,3,3,4,5,6,5,8},{3,3,4,5,6,5,8,9},{3,4,5,6,5,8,9,13},{4,5,6,5,8,9,13,15},{5,6,5,8,9,13,15,18},{6,5,8,9,13,15,18,23},{5,8,9,13,15,18,23,30},{8,9,13,15,18,23,30,33},{9,13,15,18,23,30,33,42},{13,15,18,23,30,33,42,53},{15,18,23,30,33,42,53,64},{18,23,30,33,42,53,64,75},{23,30,33,42,53,64,75,96}}
(Join[Array[a,7],{-1}].#)&/@%74
{-8+2a[1]+3a[2]+3a[3]+4a[4]+5a[5]+6a[6]+5a[7],-9+3a[1]+3a[2]+4a[3]+5a[4]+6a[5]+5a[6]+8a[7],-13+3a[1]+4a[2]+5a[3]+6a[4]+5a[5]+8a[6]+9a[7],-15+4a[1]+5a[2]+6a[3]+5a[4]+8a[5]+9a[6]+13a[7],-18+5a[1]+6a[2]+5a[3]+8a[4]+9a[5]+13a[6]+15a[7],-23+6a[1]+5a[2]+8a[3]+9a[4]+13a[5]+15a[6]+18a[7],-30+5a[1]+8a[2]+9a[3]+13a[4]+15a[5]+18a[6]+23a[7],-33+8a[1]+9a[2]+13a[3]+15a[4]+18a[5]+23a[6]+30a[7],-42+9a[1]+13a[2]+15a[3]+18a[4]+23a[5]+30a[6]+33a[7],-53+13a[1]+15a[2]+18a[3]+23a[4]+30a[5]+33a[6]+42a[7],-64+15a[1]+18a[2]+23a[3]+30a[4]+33a[5]+42a[6]+53a[7],-75+18a[1]+23a[2]+30a[3]+33a[4]+42a[5]+53a[6]+64a[7],-96+23a[1]+30a[2]+33a[3]+42a[4]+53a[5]+64a[6]+75a[7]}
#==0&/@%
{-8+2a[1]+3a[2]+3a[3]+4a[4]+5a[5]+6a[6]+5a[7]==0,-9+3a[1]+3a[2]+4a[3]+5a[4]+6a[5]+5a[6]+8a[7]==0,-13+3a[1]+4a[2]+5a[3]+6a[4]+5a[5]+8a[6]+9a[7]==0,-15+4a[1]+5a[2]+6a[3]+5a[4]+8a[5]+9a[6]+13a[7]==0,-18+5a[1]+6a[2]+5a[3]+8a[4]+9a[5]+13a[6]+15a[7]==0,-23+6a[1]+5a[2]+8a[3]+9a[4]+13a[5]+15a[6]+18a[7]==0,-30+5a[1]+8a[2]+9a[3]+13a[4]+15a[5]+18a[6]+23a[7]==0,-33+8a[1]+9a[2]+13a[3]+15a[4]+18a[5]+23a[6]+30a[7]==0,-42+9a[1]+13a[2]+15a[3]+18a[4]+23a[5]+30a[6]+33a[7]==0,-53+13a[1]+15a[2]+18a[3]+23a[4]+30a[5]+33a[6]+42a[7]==0,-64+15a[1]+18a[2]+23a[3]+30a[4]+33a[5]+42a[6]+53a[7]==0,-75+18a[1]+23a[2]+30a[3]+33a[4]+42a[5]+53a[6]+64a[7]==0,-96+23a[1]+30a[2]+33a[3]+42a[4]+53a[5]+64a[6]+75a[7]==0}
Solve[%77,Array[a,7]]
{}