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# Name:Treasa Bency Biju Jose
# Date: 10-06-2020
# Assignment #9
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1. (i) f := 1/(-4*x^3 + 3*y^3 + x*y - x - 8*y + 1);
                                 1                
             f := --------------------------------
                      3      3                    
                  -4 x  + 3 y  + x y - x - 8 y + 1


DiagSeq2(f, x, y, 50)[51];
338182851684193938056791676834675959453047728598423376623947591203531638311202321

(ii) DiagWalks3D({[1, 3, 3], [1, 3, 9], [8, 1, 1]}, 30);
[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 

  0, 0, 36, 0, 0, 0, 0, 0, 0, 0]


nops(%);
                               41

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2. 
with(gfun);
Words := proc(a, S) local T, i; 
T := []; 
for i to nops(S) do 
  if has(S, S[i]/a) then 
    T := [op(T), S[i]]; 
  end if; 
end do; 
factor(guessgf(T, x)); 
end proc;

S2 := {1, 2, 3, 4, 5, 6, 7, 8, 9, 10};
Z := Words(2, S2);
             S2 := {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

                   Z := [2/(x - 1)^2, ogf]
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3.
with(combinat);
with(gfun);
ResComps := proc(S, x) local i, j, k, F, B; 
F := []; 
for i to nops(S) do 
  B := {seq(nops(composition(S[i], j)), j = 0 .. S[i])}; 
end do; 
for k to nops(B) do 
  if not has(S, B[k]) then 
    F := [op(F), B[k]]; 
  end if; 
end do; 
factor(guessgf(F, x)); 
end proc;

R := {2, 4, 6, 8, 10};
                     R := {2, 4, 6, 8, 10}

ResComps(R, x);
                      [x*(x + 1)^9, ogf]

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4. (i) Not on OEIS
T := DiagWalks2D({[0, 1], [1, 0]}, 40);
T := [1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, 184756, 

  705432, 2704156, 10400600, 40116600, 155117520, 601080390, 2333606220, 9075135300, 35345263800, 

  137846528820, 538257874440, 2104098963720, 8233430727600, 32247603683100,  126410606437752, 

  495918532948104, 1946939425648112, 7648690600760440, 30067266499541040, 118264581564861424, 

  465428353255261088, 1832624140942590534, 7219428434016265740, 28453041475240576740, 

  112186277816662845432, 442512540276836779204, 1746130564335626209832, 6892620648693261354600, 

  27217014869199032015600, 107507208733336176461620]


gfun[listtorec](T, a(n));
[{(-n - 2) a(n + 2) + (6 + 4 n) a(n + 1), a(0) = 1, a(1) = 2}, ogf]

(ii) Not on OEIS
T := DiagWalks2D({[0, 1], [1, 0], [1, 1]}, 40);
T := [1, 3, 13, 63, 321, 1683, 8989, 48639, 265729, 1462563, 

  8097453, 45046719, 251595969, 1409933619, 7923848253, 44642381823, 252055236609, 1425834724419, 

  8079317057869, 45849429914943, 260543813797441, 1482376214227923, 8443414161166173, 

  48141245001931263, 274738209148561921, 1569245074591690083, 8970232353223635949, 

  51313576749006450879, 293733710358893793729, 1682471873186160624243, 9642641465118083682429, 

  55294491352291112747007, 317241780630136241094657, 1820991621200098527441027, 

  10457362855894862001750669, 60078868458555230983696959, 345299757825442889707393857, 

  1985346154034162284608274707, 11419126147845924397833900957, 65701922725618214591910684159, 

  378150244155138145169182750209]


gfun[listtorec](T, a(n));
[{(-n - 3) a(n + 3) + (15 + 6 n) a(n + 2) + (-n - 2) a(n + 1), 

  a(0) = 1, a(1) = 3, a(2) = 13}, ogf]

(iii) Not on OEIS
T := DiagWalks2D({[0, 1], [0, 2], [1, 0], [2, 0]}, 40);
T := [1, 2, 14, 84, 556, 3736, 25612, 177688, 1244398, 8777612, 

  62271384, 443847648, 3175924636, 22799963576, 164142004184, 1184574592592, 8567000931404, 

  62073936511496, 450518481039956, 3274628801768744, 23833760489660324, 173679413875623368, 

  1267013689048017584, 9252299435205985664, 67626504432024377756, 494710324956646794296, 

  3621791112234327295592, 26534383313499716907504, 194529413506239838951024, 

  1427026630616364232856416, 10474450985957996054150576, 76924819005707350396260192, 

  565226994478396276874021358, 4155153128570549776415060268, 30559526512591226909700271992, 

  224848729555165636463415722784, 1655038891063501813947452278468, 12186822164518997043970184661512, 

  89769447797439589097829820560392, 661476500151656115107632518838128, 

  4875741284021393250978952096137064]


gfun[listtorec](T, a(n));
[{(352*n^3 + 2768*n^2 + 7088*n + 5920)*a(n + 1) + (-484*n^3 - 4048*n^2 - 11184*n 

- 10204)*a(n + 2) + (-352*n^3 - 3120*n^2 - 9114*n - 8766)*a(n + 3) + (55*n^3 + 515*n^2 

+ 1570*n + 1560)*a(n + 4), a(0) = 1, a(1) = 2, a(2) = 14, a(3) = 84}, ogf]

(iv) Not on OEIS
T := DiagWalks2D({[0, 1], [0, 2], [1, 0], [1, 1], [2, 0], [2, 2]}, 40);
T := [1, 3, 22, 165, 1327, 10950, 92045, 783579, 6733966, 

  58294401, 507579829, 4440544722, 39000863629, 343677908223, 3037104558574, 26904952725061, 
  
  238854984979423, 2124492829796598, 18927927904130617, 168888613467092895, 

  1508973226894216106, 13498652154574126523, 120886709687492946083, 1083687170568092836350, 

  9723660300694365146989, 87322023363899721467343, 784797155029928933462614, 

  7058384549327774062817985, 63525027490219905080597647, 572078727948766828614679830, 

  5154896021503193637126695629, 46475144470335949808346622587, 419221380265186873217492981134, 

  3783331574173309950816296187025, 34158702327707968075053323650645, 

  308541107632605100591602320772210, 2788040608584603973862476702488265, 

  25202873888846971990310549726981859, 227906767043212151853904258820045110, 

  2061638450059846741700344736532466833, 18655567955631232569396589476504942877]


gfun[listtorec](T, a(n));
                              FAIL

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5. (i) A number on OEIS: A000007
T := DiagWalks3D({[0, 0, 1], [0, 1, 0]}, 50);
T := [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 

  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 

  0, 0, 0, 0, 0, 0, 0, 0, 0, 0]


gfun[listtorec](T, a(n));
                  [{-a(n + 1), a(0) = 1}, ogf]


(ii) on OEIS: Could not parse search query: query too long
T := DiagWalks3D({[0, 0, 1], [0, 1, 0], [0, 1, 1], [1, 0, 1], [1, 1, 0]}, 50);
T := [1, 4, 42, 520, 7090, 102144, 1525776, 23380368, 365130810, 

  5786380600, 92774019052, 1501646797248, 24498046138384, 402329384914240, 6645072333486720, 

  110293868867458080, 1838511122725436250, 30762545845461663240, 516457722792721132500, 

  8696582711157975572400, 146835895664138204448540, 2485266624268853914959360, 

  42157397527828247726954880, 716556351190046311597161600, 12201942968211237198350281200, 
  
  208134928903175159702319457344, 3555830679926229005075061257376, 60836819329493358733292740546048, 

  1042259677840638259492432048065280, 17878405805956405560805535960396160, 

  307035478496101645580861028622654336, 5278643189711854927868638533509575744, 

  90844825001410805657489064172778794842, 1564930724096710135146829959035675638440, 

  26982512386206869455023952107027146825700, 465628368604516908867820871730799473867408, 

  8041657743459510320462629847980612873329892, 138989457278403173476481541185549849571771136, 

  2403979582946502330471619673933968184168916960, 41607939250187986849488309498865757934964797280, 

  720614911745512461611918130596030003676289336300, 12488149132570126719418439010195646125397100011920, 

  216544860562054593134160906994219210611674235418920, 

  3756996029385637316936199664140175184307538677619200, 

  65217708880091867222809412770976203671887863352995200, 

  1132692240424335329461008559322860625660096843564044800, 

  19682044848299254204780187424606161624293922138158848000, 

  342161666360328010168145138304650840941581198111307680000, 

  5950957372613238042937365323128808212313545207994095090800, 

  103544899011058176270363371934019286365853202810957930168000, 

  1802392915688776002461574283464036323570366183012829246320352]


gfun[listtorec](T, a(n));
  [{(-567*n^3 - 4104*n^2 - 9549*n - 7140)*a(n + 1) + (-1281*n^3 - 10553*n^2 - 28546*n - 

  25400)*a(n + 2) + (-672*n^3 - 6208*n^2 - 18752*n - 18400)*a(n + 3) + (42*n^3 + 430*n^2 + 
  
  1424*n + 1504)*a(n + 4), a(0) = 1, a(1) = 4, a(2) = 42, a(3) = 520}, ogf]

(iii) on OEIS: Could not parse search query: query too long
T := DiagWalks3D({[0, 0, 1], [0, 1, 0], [0, 1, 1], [1, 0, 1], [1, 1, 0], [1, 1, 1]}, 50);
T := [1, 5, 55, 749, 11251, 178835, 2949115, 49906925, 860905315, 

  15071939255, 266982872905, 4774722189275, 86070844191775, 1561948324845095, 28507384046515555, 
 
  522867506128197869, 9631571375362268515, 178094411589895650815, 3304192479145474141741, 

  61487420580006795749999, 1147310054439368264752621, 21460298038569915209597885, 

  402303281700710751346492045, 7557003072139605981578257499, 142216452439487217855645948751, 

  2680953577121013488247925591235, 50618646171303180157424116235125, 957111527948218432660657321864175, 

  18121739540270261728632231230077015, 343543557236065454833612395025716335, 

  6520361879189996271307536840076027315, 123890073917796829080870582463811238125, 

  2356385395407636535251297627384060933475, 44861516384576265010060080056738042597135, 

  854859275252481703527187912584359902505365, 16303683385055931306937559135507039289051575, 

  311190647408479503515035135447889843225653825, 5944262823858514467535608131557811648002762625, 

  113627280554332441572175257060408538344973808425, 2173522796137552287361594832873182554514084408175, 

  41603303651437038812509444164890710254979329256765, 796818898586340848384595439852318646653791885982025, 

  15270271969281638238724672178615586077051065833360775, 

  292804205106533694230148888112886916503795367402916085, 

  5617461573225955169248923401253208251816762860322509065, 

  107826322191169954469150116083770101402658959404009097505, 

  2070721317569227785240110965397887711335116126944031387125, 

  39785206672880588731577259303187378878045846931120125328475, 

  764744172323001796507565473412652088019110691985844385750095, 

  14706094191942557902358875596276810126406906869637116414501755, 

  282916070222353435645433592830184350982846989381549672223125905]


gfun[listtorec](T, a(n));
 [{(-59*n^3 - 448*n^2 - 1084*n - 848)*a(n + 1) + (-295*n^3 - 2535*n^2 - 7185*n - 6740)*a(n + 2) + 

 (-2301*n^3 - 22074*n^2 - 69941*n - 73044)*a(n + 3) + (118*n^3 + 1250*n^2 + 4336*n + 4896)*a(n + 4), 

 a(0) = 1, a(1) = 5, a(2) = 55, a(3) = 749}, ogf]

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6. 
DiagSeq4 := proc(f, x1, x2, x3, x4, N) local i; 
if subs({x1 = 0, x2 = 0, x3 = 0, x4 = 0}, denom(f)) = 0 then 
   print(`The denominator of`, f, `should have a non-zero constant term `); 
   RETURN(FAIL); 
   end if; 
[seq(coeff(taylor(coeff(taylor(coeff(taylor(coeff(taylor(f, x1 = 0, N + 1), x1, i), x2 = 0, N + 1), x2, i), 
  x3 = 0, N + 1), x3, i), x4 = 0, N + 1), x4, i), i = 0 .. N)]; 
end proc;

DiagWalks4D := proc(S, N) local s, x1, x2, x3, x4; 
if not (type(S, set) and type(N, integer) and 0 <= N) then 
   print(`Bad input`); 
   RETURN(FAIL); 
   end if; 
DiagSeq4(1/(1 - add(x1^s[1]*x2^s[2]*x3^s[3]*x4^s[4], s in S)), x1, x2, x3, x4, N); 
end proc;

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