#OK to post homework
#Kent Mei, 11/1/20, Assignment 15

#---------------------------------
#Part 1

RoG:=proc(k,n) local V,E,i,j,T1,v,Neighs,Moves,m,pt:

V:=[seq(seq(seq([i,j],j=1..n),i=1..k))]:

for i from 1 to nops(V) do
	T1[V[i]]:=i:
od:

E:=[]:

for i from 1 to nops(V) do
	pt:=V[i]:


	Moves:={seq([j,0],j = -k..k)} union {seq([0,j],j = -n..n)}:

	Moves:= Moves minus {[0,0]}:

	Neighs:={seq(pt+m,m in Moves)}:

	Neighs:= Neighs intersect convert(V,set):

	E:=[op(E),{seq(T1[v],v in Neighs)}]:

od:

E,V:

end:

#Sequence from n = 1..6
#[2,6,96,3132,252240,36307440]

#It does not seem to be in the OEIS.


#---------------------------------
#Part 2

#NuW([40,40],{[1,0],[0,1],[1,1],[2,2]}) 
#2382564832244243056285491057263
#is the number of ways of walking from [0,0] to [40,40] with atomic steps {[1,0],[0,1],[1,1],[2,2]}.

#NuGW([40,40],{[1,0],[0,1],[1,1],[2,2]})
#89322096703094945357683861273
#is the number of those that never go above x = y.


#[seq(NuW([n,n,n],{[1,0,0],[0,1,0],[0,0,1],[1,1,1]}), n = 1..20)]
#[7, 115, 2371, 54091, 1307377, 32803219, 844910395, 22188235867, 591446519797, 15953338537885, 434479441772845, 11927609772412075, 329653844941016785, 9163407745486783435, 255982736410338609931, 7181987671728091545787, 202271071826031620236525, 5715984422606794841997001, 162016571360163769411597081, 4604748196249289767697705221]

#[seq(NuGW([n,n,n],{[1,0,0],[0,1,0],[0,0,1],[1,1,1]}), n = 1..20)]
#[2, 10, 88, 1043, 14778, 236001, 4107925, 76314975, 1491934038, 30389576308, 640286048416, 13877540824735, 308102204007536, 6983346070924707, 161156356282624227, 3778249609096250059, 89826197363219012470, 2162338803354415120414, 52637415804379149938876, 1294313658632145337351381]