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RGraphAdd[{a_,b_,c_},init_List,max_,id_:False,labs_:True]:=Graph[DirectedEdge[#[[1,1]],#[[2,1]],#[[1,2]]]&/@Last[Reap[Module[{f},f[n_]:=f[n]=a+f[Sow[n-bf[If[id,Sow[n-c,(n->2)],n-c]],(n->1)]];(f[#[[1]]]=#[[2]])&/@init;Table[f[n],{n,max}]],_,Rule]],GraphLayout->"LayeredDigraphEmbedding",VertexLabels->If[labs,Placed["Name",Center],None],VertexSize->If[labs,.7,Automatic],VertexStyle->Directive[EdgeForm[Lighter[Blue,.5]],Lighter[Blue,.9]],EdgeStyle->{DirectedEdge[_,_,1]->Blue,DirectedEdge[_,_,2]->Red}]
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RGraphAddOne[{a_,b_,c_},init_List,max_,id_:False,labs_:True]:=Graph[DirectedEdge[#[[1,1]],#[[2,1]],#[[1,2]]]&/@Last[Reap[Module[{f},f[n_]:=f[n]=a+f[Sow[n-bf[If[id,Sow[n-c,(n->2)],n-c]],(n->1)]];(f[#[[1]]]=#[[2]])&/@init;f[max]],_,Rule]],GraphLayout->"LayeredDigraphEmbedding",VertexLabels->If[labs,Placed["Name",Center],None],VertexSize->If[labs,.7,Automatic],VertexStyle->Directive[EdgeForm[Lighter[Blue,.5]],Lighter[Blue,.9]],EdgeStyle->{DirectedEdge[_,_,1]->Blue,DirectedEdge[_,_,2]->Directive[Dotted,Red]}]
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RSequenceAdd[{a_,b_,c_},init_List,max_]:=Last[Reap[Module[{f},f[n_]:=f[n]=a+f[Sow[n-bf[Sow[n-c,n->2]],n->1]];(f[#[[1]]]=#[[2]])&/@init;f[max]],_,q]]
In[]:=
RValuesAdd[{a_,b_,c_},init_List,max_]:=Module[{f},f[n_]:=f[n]=a+f[n-bf[n-c]];(f[#[[1]]]=#[[2]])&/@init;Table[f[n],{n,max}]]
In[]:=
RValuesAddOne[{a_,b_,c_},init_List,max_]:=Module[{f},f[n_]:=f[n]=a+f[n-bf[n-c]];(f[#[[1]]]=#[[2]])&/@init;f[max]]
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RNestAdd[{a_,b_,c_},init_List,max_,lim_]:=With[{rules=Flatten[{(f[#[[1]]]->#[[2]])&/@init,f[n_Integer]->a+f[Sow[n-bf[Sow[n-c,n->2]],n->1]]}]},FixedPointList[#/.rules&,f[max],lim]]
f[n]==1+f[n-f[n-3]]
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RGraphAdd[{1,1,3},{n_/;n<1->1},40,True,True]
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RGraphAddOne[{1,1,3},{n_/;n<1->1},10,True,True]
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Table[RGraphAddOne[{1,1,3},{n_/;n<1->1},m,True,True],{m,8}]
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Table[RGraphAddOne[{1,1,3},{n_/;n<1->1},m,True,False],{m,20}]
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[[[ Look at vertex stratify to get array of values “found so far” ]]]
Scanning Rules
Scanning Rules
Zero Initial Conditions
Zero Initial Conditions
How Far Back Do They Look?
How Far Back Do They Look?
The case b=1 has limited lookback because it goes one step at a time........
The case b=1 has limited lookback because it goes one step at a time........
The Add-Add Case
The Add-Add Case
TraceGraph and Call Graphs (uncached)
TraceGraph and Call Graphs (uncached)
Studying 3,1,2
Studying 3,1,2
Note these are all fairly small position values .... i.e. we did not capture the large values....
i.e. the s-th maximum is just 3 s + 1
This is the value at the s-th maximum....
Different init
Different init
This is a failure to halt.....
General Questions
General Questions
If f[n] always less than n ? Or at least asymptotically.
Universality
Universality
Can we set up values for f[n] with n < 1 so that f[m] with m > XXX corresponds to any possible function?
E.g. if the negative values are {-1,7,8,1,2,4,2} then f[m] will Boole[PrimeQ[m]] (let’s say)