> #Weiji Zheng, 11/01/2020, Assignment15
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> #OK TO POST HOMEWORK
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> 
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> 
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> 
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> #Q1
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> 
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> RoG := proc(k,n) local V,E,i,j,T1,v,Neighs,Moves,m,pt,s1,s2,s3,s4,a,b:
> 
> #The set the list of  vertices in the k by n board (using matrix indexing)
> V:=[seq(seq(seq([i,j],j=1..n),i=1..k))]:
> 
> #T1 is a labelling of the vertices in V
> 
> for i from 1 to nops(V) do
> T1[V[i]]:=i:
> od:
> 
> #E is a list such that its i-th entry is the set of neighbors of V[i] using the above-indexing
> 
> E:=[]:
> 
> for i from 1 to nops(V) do
>  pt:=V[i]:
> 
> #Moves:={[1,0],[-1,0],[0,1],[0,-1],[1,1],[-1,-1],[1,-1],[-1,1]}:

> s1 := seq([a, 0], a = 1 .. k):
> 
> s2 := seq([-a, 0], a = 1 .. k):
> 
> s3 := seq([0, b], b = 1 .. n):
> 
> s4 := seq([0, -b], b = 1 .. n):
> 
> Moves := {s1,s2,s3,s4}:
> #There are up to 8 neighbors of any vertex 
> Neighs:={seq(pt+m,m in Moves)}:
> 
> #But many of them fall off the board, 
> Neighs:=Neighs intersect convert(V,set):
> 
> #We append to the "list of neighbors" the set of neighbors of the current vertex BUT in terms of their "ids" (given by T1)
> #getting a description of the graph in "cannical form"
> E:=[op(E),{seq(T1[v],v in Neighs)}]:
> od:
> 
> #We return the graph in "canonical form" where the vertices (members of V), are labelled by positive inetegers from 1 to nops(V)
> #together with the "dictionary"
> E,V:
> end:
> 
;
> #seq(SAWnu(RoG(3,i)), i=1..6) = 2, 6, 96, 3132, 252240, 36307440
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> #NOT FOUND IN OEIS
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> 
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> #Q2
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> #1.
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> #USE SeqW
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> NuW:=proc(Pt,A) local a:
> option remember:
> 
> if Pt[1]<0 or Pt[2]<0 then
>  RETURN(0):
> elif Pt=[0,0] then
>  RETURN(1):
> else
> add(NuW(Pt-a,A),a in A):
> fi:
> end:
> SeqW:=proc(A,K) local i:
> [seq(NuW([i,i],A),i=1..K)]:
> end
SeqW := proc (A, K) local i; [seq(NuW([i, i], A), i = 1 .. K)] end proc
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> SeqW({[0, 1], [1, 0], [1, 1], [2, 2]}, 40)
[3, 14, 71, 379, 2082, 11651, 66051, 378064, 2180037, 12644861, 73695358, 431209313, 2531556197, 14904832196, 87970766447, 520337606401, 3083584244460, 18304476242735, 108820740004749, 647817646760368, 3861215365595659, 23039691494489015, 137615812845579390, 822738993163308691, 4922949521986023963, 29480139394909356044, 176664398792213185409, 1059402929496079389631, 6356902648489583020308, 38166637970127629445209, 229276573792993574603787, 1378027658578343598968422, 8286384291210047832045023, 49850407222197537838691609, 300025068023888975708739560, 1806430285705003018542769435, 10880533315007764522316552713, 65559512416370766384694103782, 395157036800184178672318869369, 2382564832244243056285491057263]
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> #2382564832244243056285491057263
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> 
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> #2.
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> #USE SeqGW
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> NuGW:=proc(Pt,A) local a:
> option remember:
> 
> if (Pt[1]<0 or Pt[2]<0 or Pt[1]<Pt[2]) then
>  RETURN(0):
> elif Pt=[0,0] then
>  RETURN(1):
> else
> add(NuGW(Pt-a,A),a in A):
> fi:
> end:

> SeqGW:=proc(A,K) local i:
> [seq(NuGW([i,i],A),i=1..K)]:
> end:
> SeqGW({[0, 1], [1, 0], [1, 1], [2, 2]}, 40)
[2, 7, 27, 116, 532, 2554, 12675, 64507, 334836, 1765833, 9434779, 50962640, 277839361, 1526834471, 8448751385, 47035469902, 263260232668, 1480527858436, 8361881707770, 47409359120684, 269736194796537, 1539547726712437, 8812663513352489, 50579825742416942, 291012706492224315, 1678146650028389023, 9697482307486861125, 56148049483650883418, 325685956663306169704, 1892343209238706561552, 11012606961635116334631, 64184027732131247229285, 374603199042488986947084, 2189213854778309191457243, 12809866709764967142650533, 75043134255268720569687608, 440108608055424139245905601, 2583842686807144572684013803, 15184702028756125034970193936, 89322096703094945357683861273]
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> #89322096703094945357683861273
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> 
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> #3.
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> #USE THE NEW SeqW
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> NuW:=proc(Pt,A) local a:
> option remember:
> 
> if Pt[1]<0 or Pt[2]<0 or Pt[3]<0 then
>  RETURN(0):
> elif Pt=[0,0,0] then
>  RETURN(1):
> else
> add(NuW(Pt-a,A),a in A):
> fi:
> end:
> SeqW:=proc(A,K) local i:
> [seq(NuW([i,i,i],A),i=1..K)]:
> end:
> 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]
;
> 
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> 
;
> #4.
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> #USE THE NEW SeqGW
;
> NuGW:=proc(Pt,A) local a:
> option remember:
> 
> if (Pt[1]<0 or Pt[2]<0 or Pt[3]<0 or Pt[1]<Pt[2] or Pt[2]<Pt[3]) then
>  RETURN(0):
> elif Pt=[0,0,0] then
>  RETURN(1):
> else
> add(NuGW(Pt-a,A),a in A):
> fi:
> end:
> SeqGW:=proc(A,K) local i:
> [seq(NuGW([i,i,i],A),i=1..K)]:
> end:
> 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]
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> 
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