#OK to post hw
#Michael Yen, 11/8/20, Assignment 17
#Lecture 17: Due Nov. 8, 9:00pm. Email ShaloshBEKhad@gmail.com an attachment

#hw17FirstLast.txt

#Indicate whether it is OK to post

#1) Use the appropriate procedure in ComboProject1.txt to find the number of ways a King can travel on a 3 by 50 chessboard, return to the starting square and visit each of the 150 squares exactly once.

SAWnu(KiG(3,50)[1])

#I wasn't able to get an answer back from this one.

#2) Use the appropriate procedures in ComboProject2.txt to find the number of ways of walking from [0,0] to [3000,3000] using, as atomic steps {[1,0],[0,1],[1,1],[1,2],[2,1]}, all the while staying in the region x ≥ y
#Don't do it directly! Use Findrec followed by SeqFromRec.

l:=[seq(NuGW([i,i],{[1,0],[0,1],[1,1],[1,2],[2,1]}),i=1..100)]:
ope:=Findrec(l,n,N,6);
SeqFromRec(ope,n,N,[l[1],l[2],l[3],l[4]],3000)[3000];

#95169091361304497661016734197678134770349822930308569276916914467565347871540683291808752282824707570651985872820372363152563131551499362536825304711559361666563103386330829999710178259606336618359856705840905896307823039694186915923739801088785552537207794661641719063082723104319582945468736368770654686751378287041101572685985713084393957598627298986861938927166053927103431921104126027579703091359302890708953724730599070838409654723308763002613165079306548111632722909218562889533100675786353470112016022196477554880490589499289089162582632715079250461763612854611196880347954440249628671174622179674903282050871247325146410790010719334253540172397663997792921830518354359522669329363082614972729965686428855219511297199827548476675004412199720715365099792501311123143368483881397876775712439421054798831717981977936622337062303624546576898187959341646575490702549296915200857634048143923818701102393144599106994024665053648913638391157955194662231996921003344586526221812096107389514125990119451880185515845934698037055104486896573666844857573880742852639391722910231692794433520126339340352663869698430014805776170895308465157999383126154136534892638175457579256315002808693778321645459807540062485196275478961352224073937381392596094278147716000344452101216213718218846995422403246683326563202994807095282839244324106363086847480478463730345215985413064515655006280716371594297868314185041474747026816679035293127055797403755491278529530018110422188400807126145601136460544996907253108026710738133221241693706980842171873604900269856459541793234702167937163679886476596787643003424261664266808615823066346661804622908642236679528725542671747404314012249795021732170691008575336656050986306648593100164229972842611760785390415331373519508784472911981975537222743422312589751576257948909815276363791495173342977855094118355700016556226919615428250897733703209967582718607928464069524222159738340806195626241390924709235357374015016254839959209084054394569432471432576684522506998779788376200443126782132255047582891261666245361634285199523372050333572627938881441014457739772722775254541205569963083330946306737155223437987340685548656285567503838100825537658312256841838804244651258118811651131924827871017999511708744102921789247462817980894438730298698541980861879749020341945718568693083738317799328488585909649259726166330681988951268751122991503360075439622584181083778123619416313964846558075294830227299727374990124401764015434769152111199549890678059731046462531157554362228562028418125254044367001279506736510680059995930940034

#3) Use the appropriate procedures in ComboProject3.txt to find the number of ways of walking from [0,0,0] to [3000,3000,3000] using, as atomic steps {[1,0,0],[0,1,0],[0,0,1],[1,1,1]}, all the while staying in the region x ≥ y ≥ z

#Don't do it directly! Use Findrec followed by SeqFromRec.

l:=[seq(NuGW([i,i,i],{[1,0,0],[0,1,0],[0,0,1],[1,1,1]}),i=1..100)]:
ope:=Findrec(l,n,N,5);
SeqFromRec(ope,n,N,[l[1],l[2],l[3]],3000)[3000];

#1055961597172638192205714533827955182069330710185887123715638338751912356623868934358954829265893265497184019516455497946501358614406592180916562995571435822638011864699948413518735379729837425167828197932773091617214603560704076763403058008870590210530506834889187670236069663000563403909291518139471141193933668569097230791592378327765453206880338367141465884774689273332974613327790670133864770839786907616252716172612810392322684357026318977543761631185892041752872792707472523339315464878092600122776510487117547611700101814824458512731561188677556925181977017823995965449343853533549328389007002003812727585993220294254740294866113923151479672429490481644088528411109634826771309376893827003349243068183476873533935594331988585675337762038394870883132602159173883922347999944078821728163370937052619155590568391069063317592509955598788483466418234408407388202299036233322379192615075021698393212868787733569268229184628289565589549741584574327892919908881010751517221722381012834556556824848577221227684137830816101020327876526681725444453682864636555747673248741474708131366459673977930419195224272687150280798146593067544755578201466060570093254935347120068461604351415848203621522648676607169207523056123186434484713066312780810175397901692245443312244602960976968454897460424230265560584944905249327347768975517831568652679595607920084575082584347314212810007985240690964634053969181170611215275387792197501045392327155456594027530404389193373166529959026406585893677752993907919001817631111686530612365744087948657981705511840245825626621270255357615436919191022104201504251202188647227781935143967146969540653262038857149021960306229530963054026491708200678862903006329234012195611754127295080517138419193838585920213543419469950079348946755215094694137847178186210286204470627994554250221170678469119109643939053546724329890548255475005106738694859949241621137720195040487766275618591781040165985869235563695957345264145252201054929162788252799455694036338904769031986390960043941243280986491172713827805499916660157385659508981509849147715973547457229029295526584351410281813731593170470817309110789876049747009019009949757200409333866914797127029381219525171081649367682900079983068290393154404118264018185364905606965589045505005572066630122121150375280811289992817027578011638093603433244372443437925332947876282657417807723726415290604403918202072098770116006990175350820492314867796387685661273702215556922453698175420527665425961486623775069041082819614980223231440570232620689816525157996858630797825675874156474646732065652602821773250214911095523839301872173514572945534777185829446594458412939559790823335335988943249612096741437113159733496107309438749517155816113611466181606413786593079760286476932502122540382520346014562754440684858961033148757521944516188781560550967342223679376726927253300939809903760523174681426976332580145397706760699651938934492314536611971407185558853691943983900622101911869433034988901671398749303046154807811343347079647262775389547959397251809798861843882195477113660504281821150366169039324164617598327207275003383926645719618807107118775499837007692943421107557066566151632494226460059669482223627000755320102291265532871104516620408021503469564960744840527318501733587855812003819890890023893498130260498951286347694871337136866334557342356711006118238958427510975755322669497692071461010165591733674071285408880586240632956831541705216908849696753484260512346231146506771015879263648500135313751979294236700008885800509693120930841769817301480352739004267277612475636269175159040483895116762728473969354479429135527425822803355504559236868842150825399906121046596018792183036698610028246005663710973350043476200695248674446173296357513573521498183102888985070332615743303290931992949087204064596760980551855974772314932770996526784911225488608910212901539625649853695492476183650987182998517921557222998098782725632700696246594120460326038060483755479352197136229776118789296545919836632814452748621216052104394615335775039904853067848468754958862041044060581067912446471855014833235066207766370613368257308071954943653479180111042281881188887660434936081241799850788470218012237849400298660425782403148718872653901005280466525095291577169601084339235634955604898374949141660578599870423680499360451073008559080164788699736386609233499025062855391709957273487073386537359165683384170380942428413622358059424469862102618894310733782269292492850029973291

#4) Use the appropriate procedures in ComboProject4.txt to find
Sum(binomial(3000,k)^2*binomial(3000+k,k)^2,k=0..3000)

#Don't do it directly! Use Z and then use SeqFromRec.

ope:=Z(binomial(n,k)^2*binomial(n+k,k)^2,k,n,N):
ope2:=subs(n=3000,ope);
SeqFromRec(ope2,n,N,[1,81054009000001],3000)[3000];

#Didn't get an answer back from Maple.

#5) Use the appropriate procedures in ComboProject5.txt to find the number of 3 by 101 TicTacToe boards that ended in a tie (i.e. had 151 X's and 150 O'x and no three consectutive Xs, or O's horizontally, vertically, or diagonally.

OddTTT3(50)[50]

#1290808418002135615116

#6) Use the appropriate procedures in ComboProject6.txt to approxinate the average degree of an induced graph if you take 1000 random choices of 257 vetrices in Bn(9). Of coure it changses from run to run, but it should be close.

#SimuAvDegree(Bn(9)[1],257,1000);

[4507.587549, 4503.079963]

#Average degree is 4507.587549.

#7) Use the appropriate procedures in ComboProject7.txt to find the coefficient of x2000 y2000 in the bi-variate Taylor series of
1/(1-4*x-5*y+11*x*y)

l:=DiagSeq2(1/(1-4*x-5*y+11*x*y),x,y,100):
ope:=Findrec(l,n,N,3);
SeqFromRec(ope,n,N,[l[1],l[2]],2000)[2000];

#Don't do it directly!

#773646859915755345080592880711987868114000067411267711089645465297302185782140437651737245647882206781438666359768916791715657797616253271074975816775510891610817784007689815061811713118793808132127169156531956632210202931839862195808509640930194072475736797648958419891588400228778302107480391146576468783927091978800730725381748927502387454205436928125198975511378423794707365823379926986188749731089986500669917397396616873971504095084806867185794148201994249569708224664939358898156705285766137705808523882246825460865055845849608101602554124421655543281954942869347763265534288898345102646980249705245750411149901021906533804652856542653764498677365214438615665150615517343477647558880827613427285885357921841100046634238461334120207717747344306854642161789096422806822477126080459383319139414084891874813390575035056820274820249163621592543772865926118882829787233403086035287238779765054941813725170004414582374135508817403257313682626312641850119576783617936793647393405068210227926377575854139336013624990425865640946215471320032948282468940324846914561138775497162509724260118084959559782824445762245632299836411179392775648099505004754289691484084437878335072556789083071128828642026102198988713371433916086729092810868873365284186949520302620916959510360154686720720849426575947536973822089840926819039524739154650307471233566116603524983355463110367522269735546877184343991825037545531189756117352626296189893351782094577197526461536355028695315424139758095138975823719895497596876668131934199439173229485368636994380428110961997982663512993479753274524564588897844411090731278082694716259642257912209226715162964982766205644375213279962095754671175676851278122336924210409928907451885674299545970520895098122438167730023956375082767841919536492659592553484758514186588962566254898054642148157529356160985269637101035563320027681988296575398258581899930433851537298396125459746563645498518344619371613618855757920153091128439831385651061713354178518359400614334182811551611306246590748272172228421650703852569210285063688014891546768428033707913809454151141697846121976041016161198422075270533277284589452027578075668399593831720510523495731158754673764413760975340577700256181793723268102737700377434630624790249458797868911788044600805508485574241229253198181585117115731176981772669466598392654721571804299532408697203358580912723946227908232155338588428127174145078183468669908828952557055163469676170155844087464068982383237074344331350973287832545815695040580098426592501971544206619030825683346801296280335524080438127590769794668939223630265567779224717497406424337680111319658968840828783394398066924082596974045522418906009355130535708234172204495769135348785825545908155547519817718720637411445747313149486470464529574488009448609563739552269119003138524190110775062801488158287029560619172943034584983763324488306289724561326554924726583458117754924933611552675041469797148844028192904120049411967702119285010310450399474899988993586144695013187426300062274614261232855180913589064620769172807279547595522263494030111333199971899957574277407800812548961437147806523394123808961636877041743608198426614043951238220331970986440053143097092028066307341624392350207383551740144663484185505423935912306650375320554777485131772136130297787225160458443358096735851606998995447761379827375728762226123218580834402190682606068437153306290925913239883594710884034531237609827555387947652206110770058576176763122989223205695856623992082656117211112586521163319820736781340880001