NeighborWeightedLists[g_,i0_,n_]:=NestList[NW0[g,#]&,{{i0,1}},n]
NW0[g_,list_]:=({#[[1,1]],Apply[Plus,Last[#]]}&[Transpose[#]])&/@Split[Sort[Flatten[(Function[u,{u,Last[#]}]/@Replace[First[#],g])&/@list,1]],First[#1]==First[#2]&]
NeighborWeights[g_,i0_,n_]:=Map[Sort[Last/@#]&,NeighborWeightedLists[g,i0,n]]
NeighborWeights[TwoD[8],1,6]
{{1},{1,1,1},{1,1,1,1,1,1,3},{1,1,1,1,1,1,2,2,2,5,5,5},{1,1,1,1,2,2,2,2,2,2,2,8,8,8,8,8,8,15},{1,1,1,1,2,3,3,3,3,4,6,11,11,11,11,12,12,18,18,18,31,31,31},{1,1,1,1,3,3,3,3,6,6,15,15,15,15,24,24,27,27,27,27,30,60,60,60,60,61,61,93}}
NeighborWeightedLists[TwoD[8],1,6]
{{{1,1}},{{2,1},{9,1},{57,1}},{{1,3},{10,1},{16,1},{17,1},{49,1},{58,1},{64,1}},{{2,5},{8,2},{9,5},{11,1},{18,2},{24,1},{25,1},{41,1},{50,2},{56,1},{57,5},{59,1}},{{1,15},{3,2},{7,2},{10,8},{16,8},{17,8},{19,1},{23,1},{26,2},{32,2},{33,2},{42,2},{48,2},{49,8},{51,1},{55,1},{58,8},{64,8}},{{2,31},{4,2},{8,18},{9,31},{11,11},{15,3},{18,18},{20,1},{24,11},{25,12},{27,3},{31,1},{34,6},{40,4},{41,12},{43,3},{47,1},{50,18},{52,1},{56,11},{57,31},{59,11},{63,3}},{{1,93},{3,24},{7,24},{10,60},{12,3},{14,3},{16,60},{17,61},{19,15},{23,15},{26,27},{28,1},{30,1},{32,27},{33,30},{35,6},{39,6},{42,27},{44,1},{46,1},{48,27},{49,61},{51,15},{55,15},{58,60},{60,3},{62,3},{64,60}}}
ShellNeighborWeightedLists[TwoD[6],1,6]
{{{1,1}},{{2,1},{7,1},{31,1}},{{8,1},{12,1},{13,1},{25,1},{32,1},{36,1}},{{6,2},{9,1},{14,2},{18,1},{19,2},{26,2},{30,1},{33,1}},{{3,2},{5,2},{15,1},{17,1},{20,4},{24,4},{27,1},{29,1}},{{4,2},{11,3},{16,1},{21,6},{23,2},{28,1},{35,3}},{{10,6},{22,4},{34,6}}}
ShellNeighborWeights[TwoD[6],1,6]
{{1},{1,1,1},{1,1,1,1,1,1},{1,1,1,1,2,2,2,2},{1,1,1,1,2,2,4,4},{1,1,2,2,3,3,6},{4,6,6}}
ShellNeighborWeights[TwoD[20],1,6]
{{1},{1,1,1},{1,1,1,1,1,1},{1,1,1,1,1,1,2,2,2},{1,1,1,1,1,1,2,2,2,2,2,2},{1,1,1,1,1,1,2,2,2,3,3,3,3,3,3},{1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3}}
ShellNeighborWeights[TwoD[40],1,10]
{{1},{1,1,1},{1,1,1,1,1,1},{1,1,1,1,1,1,2,2,2},{1,1,1,1,1,1,2,2,2,2,2,2},{1,1,1,1,1,1,2,2,2,3,3,3,3,3,3},{1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3},{1,1,1,1,1,1,3,3,3,3,3,3,4,4,4,4,4,4,6,6,6},{1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6},{1,1,1,1,1,1,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,10,10,10,10,10,10},{1,1,1,1,1,1,5,5,5,5,5,5,5,5,5,5,5,5,10,10,10,10,10,10,10,10,10,10,10,10}}
Length/@%350
{1,3,6,9,12,15,18,21,24,27,30}
NeighborWeights[OneD[8],1,6]
{{1},{1,1,1},{2,2,2,3},{6,7,7,7},{20,20,20,21},{60,61,61,61},{182,182,182,183}}
NeighborWeights[OneD[40],1,6]
{{1},{1,1,1},{1,1,2,2,3},{1,1,3,3,6,6,7},{1,1,4,4,10,10,16,16,19},{1,1,5,5,15,15,30,30,45,45,51},{1,1,6,6,21,21,50,50,90,90,126,126,141}}
ShellNeighborWeights[OneD[40],1,6]
{{1},{1,1,1},{1,1,2,2},{1,1,3,3},{1,1,4,4},{1,1,5,5},{1,1,6,6}}
ShellNeighborWeights[ThreeD[8],1,6]
{{1},{1,1,1},{1,1,1,1,1,1},{1,1,1,1,1,1,1,1,1,1,1,1},{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2},{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,4},{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}}
Length/@%
{1,3,6,12,23,29,32}
NeighborWeights[TreeNetwork[4],{2},6]
{{1},{1},{1,1,1},{1,1,1,1,1,1,3},{1,1,1,1,1,1,1,1,1,1,1,1,5,5,5},{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,7,7,7,7,7,7,15},{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,9,9,9,9,9,9,9,9,9,9,9,9,29,29,29}}
ShellNeighborWeights[TreeNetwork[4],{2},6]
{{1},{1},{1,1,1},{1,1,1,1,1,1},{1,1,1,1,1,1,1,1,1,1,1,1},{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1},{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}}
Length/@%
{1,1,3,6,12,24,48}
NeighborWeights[SphereNetwork[2],{1},6]
{{1},{1},{1,1,1},{1,1,1,1,1,1,3},{1,1,1,1,1,1,2,2,2,5,5,5},{1,1,1,1,1,1,2,2,2,2,2,2,8,8,8,8,8,8,15},{1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,11,11,11,11,11,11,18,18,18,31,31,31}}
ShellNeighborWeights[SphereNetwork[2],{1},6]
{{1},{1},{1,1,1},{1,1,1,1,1,1},{1,1,1,1,1,1,2,2,2},{1,1,1,1,1,1,2,2,2,2,2,2},{1,1,1,1,1,1,2,2,2,3,3,3,3}}
NeighborWeights[RandomNetwork[100],{1},6]
{{1},{1},{1,1,1},{1,1,1,1,1,1,3},{1,1,1,1,1,1,1,1,2,2,5,5,5},{1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,7,7,7,8,8,9,15},{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,9,9,9,9,9,10,10,11,18,20,30,30,31}}
ShellNeighborWeights[RandomNetwork[100],{1},6]
{{1},{1},{1,1,1},{1,1,1,1,1,1},{1,1,1,1,1,1,1,1,1,1},{1,1,1,1,1,1,1,1,1,1,1,1,1,1},{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}}
ShellNeighborWeights[RandomNetwork[100],{1},10]
{{1},{1},{1,1,1},{1,1,1,1,1,1},{1,1,1,1,1,1,1,1,2},{1,1,1,1,1,1,1,1,1,1,1,1,2,2,2},{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2},{1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4},{1,1,1,1,1,2,2,2,2,2,2,3,3,3,4,4},{5},{}}
ShellNeighborWeights[RandomNetwork[100],{1},10]
{{1},{1},{1,1,1},{1,1,1,1},{1,1,1,1,1,1,1,1},{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1},{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2},{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,4},{1,1,1,2,2,2,2,2,2,2,2,2,3,3,5},{2,2,3,4,7},{}}
Length/@%
{1,1,3,4,8,16,21,27,15,5,0}
Apply[Plus,NeighborWeights[OneD[40],1,6],{1}]
{1,3,9,27,81,243,729}
Apply[Plus,NeighborWeights[TwoD[8],1,6],{1}]
{1,3,9,27,81,243,729}
Apply[Plus,ShellNeighborWeights[OneD[40],1,6],{1}]
{1,3,6,8,10,12,14}
Apply[Plus,ShellNeighborWeights[TwoD[8],1,6],{1}]
{1,3,6,12,18,28,28}
Apply[Plus,ShellNeighborWeights[TwoD[14],1,10],{1}]
{1,3,6,12,18,30,42,66,88,118,124}
ShellNeighborCounts[TwoD[14],1,10]
{1,3,6,9,12,15,18,20,20,19,18}
ListPlot[%430,PlotJoined->True];
From random Voronoi:
ListPlot[{1,1,3,6,12,18,26,34,42,53,67},PlotJoined->True];