#OK to post homework
#William Wang, 11/1/2020, Assignment 17

#1.
KingToursGF(3, 50, t);
                    2 /   4      3      2    \
                 2 t  \3 t  + 4 t  + 2 t  - 2/
                 -----------------------------
                    4      3       2          
                 6 t  + 8 t  + 15 t  + 4 t - 1

coeff(taylor(%, t = 0, 150), t, 149);
 1158539046915934049714422113402236447096268243608991870109582897563283760401185844487539235734704107286551596007582669512



#2.
S := InfoGA({[0, 1], [1, 0], [1, 1], [1, 2], [2, 1]}, n, N, 50, 5, 5);
     [                                                          
     [                                                          
S := [{[0, 1], [1, 0], [1, 1], [1, 2], [2, 1]}, [2, 8, 37, 187, 
     [                                                          
     [                                                          

  1005, 5633, 32566, 192792, 1162918, 7121902, 44165560, 

  276783968, 1750207940, 11153013440, 71550898197, 461746445731, 

  2995462031745, 19523143468973, 127777132138891, 

  839452832023207, 5533837097342891, 36594085887854795, 

  242680221398481381, 1613598426851118817, 10754859380597438069, 

  71842748551816723277, 480905190308830819225, 

  3225307676284611819217, 21670117616522081549953, 

  145840679672699944094433, 983053428271497814915238, 

  6636125814739273692660888, 44859345005089675067344174, 

  303638741080314921979131686, 2057763386337603966845020668, 

  13961702500509954172092227612, 94832904877320251327289447656, 

  644811242965640714216092401476, 

  4388701626430898296019153105026, 

  29898414203468535846138390257098, 

  203868052178975450132547578090878, 

  1391299997423028635726671660812258, 

  9502676021281003270491972199172678, 

  64954415289919470625783673910590162, 

  444318286253185235104169038220769834, 

  3041505111745155334565491437675951478, 

  20834291953518436333856265143938254112, 

  142808006475300877885644385533683789544, 

  979486623322036695091659490280293252388, 

                                               2 (2 n + 3)
  6722113044436452619618838139392108932544], - -----------
                                                  n + 5   

                                  2              3       
     (11 n + 23) N   (13 n + 41) N    5 (n + 3) N     4  
   - ------------- - -------------- - ------------ + N , 
         n + 5           n + 5           n + 5           

                                                     1       
  7.070105942718174702750832392351662467997, 1, ------------ 
                                                 1.500000001 
                                                n            

  /                        n /     1.384505747   1.405966040
  |0.8253031212 7.070105943  |1. - ----------- + -----------
  |                          |          n             2     
  \                          \                       n      

                                              ]
     1.307299664   1.384210837   1.497633044\\]
   - ----------- + ----------- - -----------||]
          3             4             5     ||]
         n             n             n      //]


findrec(S[2], 1, 4, n, N);
                                             2              3     
  2 (2 n + 3)   (11 n + 23) N   (13 n + 41) N    5 (n + 3) N     4
- ----------- - ------------- - -------------- - ------------ + N 
     n + 5          n + 5           n + 5           n + 5         

SeqFromRec(findrec(S[2], 1, 4, n, N), n, N, [2, 8, 37, 187], 3000)[3000];
9516909136130449766101673419767813477034982293030856927691691446\

  756534787154068329180875228282470757[...2343 digits...]9152111\

  19954989067805973104646253115755436222856202841812525404436700\

  1279506736510680059995930940034



#3.
A := InfoGA({[0, 0, 1], [0, 1, 0], [1, 0, 0], [1, 1, 1]}, n, N, 50, 6, 5);
     [                                                          
     [                                                          
A := [{[0, 0, 1], [0, 1, 0], [1, 0, 0], [1, 1, 1]}, [2, 10, 88, 
     [                                                          
     [                                                          

  1043, 14778, 236001, 4107925, 76314975, 1491934038, 

  30389576308, 640286048416, 13877540824735, 308102204007536, 

  6983346070924707, 161156356282624227, 3778249609096250059, 

  89826197363219012470, 2162338803354415120414, 

  52637415804379149938876, 1294313658632145337351381, 

  32118082448138745067175242, 803656789698224684452667975, 

  20262539635339176715000960941, 514456553505357928725536149803, 

  13146084523635788235064716197728, 

  337929024350121818044901686404816, 

  8734691492940974137071498064995420, 

  226931194428057541769204674879130191, 

  5923987397778903817189358328568041496, 

  155335609958542073698381974113868127891, 

  4090172312867860136825480052716884724147, 

  108121976818168155620957000319917917053675, 

  2868707794254575240567639311917494731749030, 

  76377538885644568975349924729052848698617526, 

  2040172740682907191699950846063214773112927416, 

  54665344392211673342456683384055104205997175257, 

  1469025374919143636369316715736858925903810273670, 

  39586979783512218527277338099905807341274587256147, 

  1069596920795621360143277142155882112468901055822299, 

  28971834889893726116466068771262180074158013962368089, 

  786624416885179957698948378595042627165085805977597238, 

  21406496941578870323675397740377275998112856431253739940, 

  583801369714608515305502011392156725334624008192909440168, 

  15954513928133137501267139726262869284981213941344807489857, 

  436879272160793629128022091511848928647620180950721848780592, 

  11985652641165379506890719555297678340424492500261026978221269, 

  329419762705629446970296352861004019084652585012933306375828775, 

  9069697602289298897479257949144657270732090889455197964425043207, 250127021374018898282936610542536017519164050047115725607341714968, 6909145395201492293479975824651120370386436412105562027156097665318

                                           /   2           \  
       (n - 1) (n - 3) (3 n + 8)   (n + 3) \9 n  - 3 n - 16/ N
  ], - ------------------------- + ---------------------------
       (3 n + 5) (n + 5) (n + 4)    (3 n + 5) (n + 5) (n + 4) 

                 /    2            \  2       
     2 (3 n + 7) \15 n  + 65 n + 62/ N     3  
   - ---------------------------------- + N , 
         (3 n + 5) (n + 5) (n + 4)            

                                                      1       
  29.90078668849808557721893812757244878657, -4, ------------ 
                                                  3.999999966 
                                                 n            

  /                        n /     3.938320803   10.82756283
  |0.7671120615 29.90078669  |1. - ----------- + -----------
  |                          |          n             2     
  \                          \                       n      

                                              ]
     25.57892539   55.93103332   115.8277922\\]
   - ----------- + ----------- - -----------||]
          3             4             5     ||]
         n             n             n      //]


findrec(A[2], 3, 5, n, N);
                                         /   2           \  
     (n - 1) (n - 3) (3 n + 8)   (n + 3) \9 n  - 3 n - 16/ N
   - ------------------------- + ---------------------------
     (3 n + 5) (n + 5) (n + 4)    (3 n + 5) (n + 5) (n + 4) 

                    /    2            \  2     
        2 (3 n + 7) \15 n  + 65 n + 62/ N     3
      - ---------------------------------- + N 
            (3 n + 5) (n + 5) (n + 4)          


SeqFromRec(findrec(A[2], 3, 5, n, N), n, N, [2, 10, 88], 3000)[3000];
1055961597172638192205714533827955182069330710185887123715638338\

  751912356623868934358954829265893265[...4214 digits...]7273487\

  07338653735916568338417038094242841362235805942446986210261889\

  4310733782269292492850029973291



#4.
seq1 := SeqFromRec(Z(((3000 + k)!)^2/((k!)^4*((3000 - k)!)^2), k, n, N), n, N, [], 3000);
sum(seq1[i], i = 1 .. nops(seq1));
#Not sure, Z takes too long to run



#5.
OddTTT3(50)[50];
                     1290808418002135615116



#6.
(sum(AvDegIG(Bn(9)[1],{seq(Bn(9)[1][i],i=rand(1..257)}),j=1..1000))/1000;
                               2.



#7.
L := DiagSeq2(1/(11*x*y - 4*x - 5*y + 1), x, y, 100);
R1 := Findrec(L, n, N, 3);
                  121 (n + 1)   29 (2 n + 3) N    2
            R1 := ----------- - -------------- + N 
                     n + 2          n + 2          

SeqFromRec(R1, n, N, [seq(L[i], i = 1 .. 2)], 2000)[2000];
7736468599157553450805928807119878681140000674112677110896454652\

  973021857821404376517372456478822067[...3292 digits...]5387947\

  65220611077005857617676312298922320569585662399208265611721111\

  2586521163319820736781340880001