#OK to post homework
#William Wang, 10/25/2020, Assignment 15

#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 to nops(V) do
 T1[V[i]] := i:
od:
E := []:
for i to nops(V) do
 pt := V[i]:
 Moves := {[-7, 0], [-6, 0], [-5, 0], [-4, 0], [-3, 0], [-2, 0], [-1, 0], [0, -7], [0, -6], [0, -5], [0, -4], [0, -3], [0, -2], [0, -1], [0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [0, 6], [0, 7], [1, 0], [2, 0], [3, 0], [4, 0], [5, 0], [6, 0], [7, 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:

seq(nops(SAW(RoG(3, n)[1])), n = 1 .. 5);
                     2, 6, 96, 3132, 252240

#This sequence is not in the OEIS



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

#The number of ways of walking from [0,0) to [40,40] with atomic steps {[1,0],[0,1],[1,1],[2,2]} that never go above x=y
NuGW([40, 40], {[0, 1], [1, 0], [1, 1], [2, 2]});
                 89322096703094945357683861273



#3.
#The first 20 terms of the sequence enumerating walks from [0,0,0] to [n,n,n] with atomic steps {[1,0,0],[0,1,0],[0,0,1],[1,1,1]}
SeqW({[0, 0, 1], [0, 1, 0], [1, 0, 0], [1, 1, 1]}, 20);
[7, 115, 2371, 54091, 1307377, 32803219, 844910395, 22188235867, 

  591446519797, 15953338537885, 434479441772845, 

  11927609772412075, 329653844941016785, 9163407745486783435, 

  255982736410338609931, 7181987671728091545787, 

  202271071826031620236525, 5715984422606794841997001, 

  162016571360163769411597081, 4604748196249289767697705221]

#The first 20 terms of the sequence enumerating walks from [0,0,0] to [n,n,n] with atomic steps {[1,0,0],[0,1,0],[0,0,1],[1,1,1]} that stay in x>=y>=z
SeqGW({[0, 0, 1], [0, 1, 0], [1, 0, 0], [1, 1, 1]}, 20);
[2, 10, 88, 1043, 14778, 236001, 4107925, 76314975, 1491934038, 

  30389576308, 640286048416, 13877540824735, 308102204007536, 

  6983346070924707, 161156356282624227, 3778249609096250059, 

  89826197363219012470, 2162338803354415120414, 

  52637415804379149938876, 1294313658632145337351381]