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#Mathematica, 118117 bytes
If[#==1,{0,1},#&@@SortBy[(S=Select)[S[Range[0,s=#]~Tuples~2,#<#2&@@#&]Less@@#&],!FreeQ[LinearRecurrence[{1,1},#,s],s]&],Mean]]&
#Mathematica, 118 bytes
If[#==1,{0,1},#&@@SortBy[(S=Select)[S[Range[0,s=#]~Tuples~2,#<#2&@@#&],!FreeQ[LinearRecurrence[{1,1},#,s],s]&],Mean]]&
#Mathematica, 117 bytes
If[#==1,{0,1},#&@@SortBy[(S=Select)[S[Range[0,s=#]~Tuples~2,Less@@#&],!FreeQ[LinearRecurrence[{1,1},#,s],s]&],Mean]]&
#Mathematica, 148118 bytes
(a=Join@@Table[{j,i},{i,#}If[#==1,{j,0,i-1}];d=z=1;If[#==1,#&@@a#&@@SortBy[(S=Select)[S[Range[0,While[d!=#s=#]~Tuples~2,t=1;While[(s=#&@@LinearRecurrence[#<#2&@@#&],!FreeQ[LinearRecurrence[{1,1},a[[z]]#,{t++}])<=#s],d=s];z++];a[[z-1]]])&s]&],Mean]]&
#Mathematica, 148 bytes
(a=Join@@Table[{j,i},{i,#},{j,0,i-1}];d=z=1;If[#==1,#&@@a,While[d!=#,t=1;While[(s=#&@@LinearRecurrence[{1,1},a[[z]],{t++}])<=#,d=s];z++];a[[z-1]]])&
