In[]:=
dat=TableClearAll[f,g,floorTerm,kMax,a];f[n_]:=x*3^n+Sum[3^(n-1-j)*a[j],{j,0,n-1}];g[n_]:=(2f[n]-1+2^(k+1))2^(k+2);floorTerm[n_]:=(Floor[g[n]]-Floor[g[n]/2]-Floor[f[n]/2^(k+2)]);kMax[n_]:=Floor@Log@f[n];a[0]=1;a[n_]:=a[n]=2(f[n]-Sum[2^kfloorTerm[n],{k,0,kMax[n]}]);(*lookatlimitingbehaviorofratioofa[n]tox*5^n(n=10isalreadygoodenoughforlookingatthelimit)*)x,N@,{x,1,10^6,10^4};(*lookslikea[n]isalmostproportionalto5^nforlargenandanyx,althoughtheremightbeanadditionalsub-logarithmicgrowthtermastheratioisslowlygrowing*)ListPlotdat,Joined->True,Frame->True,FrameLabel->"x","[n]"
a[10]
x*5^10
lim
n->∞
a
x
x*
n
5
Out[]=