The first, 40, terms of the enumerating sequences of walks with one positive \ and one negative step each of size between 1 and, 9 that start and end at the x-axis and never go above the x-axis By Shalosh B. Ekhad In this book we will present the first, 40, terms for the sequences enumerating ALL families of walks consisting of at one negative and one positive step where the size of each step is between 1 and, 9 and that start and end at the x-axis, but never go above the x-axis -------------------------------------------- Fact No., 1 Let a(m) be the number of sequences of length, 5 m, in the alphabet, {-2, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [2, 23, 377, 7229, 151491, 3361598, 77635093, 1846620581, 44930294909, 1113015378438, 27976770344941, 711771461238122, 18293652115906958, 474274581883631615, 12388371266483017545, 325714829431573496525, 8613086428709348334675, 228925936056388155632081, 6112355595348903438948155, 163869604996054172670563730, 4409530805291138147009940577, 119053554405655805081339777916, 3224183996664038053758382236200, 87561359887231436502479212567970, 2384085494382142705840768924534711, 65066994871847852574373263376452916, 1779721375844162673264781567864062809, 48778310711916131693983042017623427225, 1339440127229098961724679335624540335066, 36845651741358843132371645279597764145919, 1015232633493863663403064931278165311426407, 28016638350988616504142110219146979406071917, 774277261749708515208592839929664033191044807, 21427404838561062531091043642063820036679833389, 593746462814296689149461136826877135951332690628, 16472501907732640366633552130602300314405354565691, 457526666710847771332143448705779885372534218160782, 12721698698191000148296283521505207982847394689460065, 354096175405843316837783350030659195321133204139875850, 9865571268349576393705796834229371307236251306950570050] ---------------------------------------- This ends Fact No., 1, that took, 0.432, seconds to generate. -------------------------------------------- Fact No., 2 Let a(m) be the number of sequences of length, 7 m, in the alphabet, {-2, 5}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [3, 76, 2803, 121637, 5782513, 291437249, 15297882929, 827402061954, 45790180469312, 2580588279994441, 147592910517101281, 8544927937132306600, 499811636639428519226, 29491983283370728013309, 1753398440591481772556798, 104933899400256659634374549, 6316334518803437568442071134, 382157894402929888870502301422, 23227867059873378295000835711897, 1417626021933575573690480062573957, 86841335288654276627784306088376579, 5337703991119630400867518436365901430, 329091133138293837644741627024856762971, 20346914629793261198471539429714087455949, 1261253541170025826081983672206908041769463, 78368238337961270215187805156530040517653795, 4880158146656704277707955417370236584894808458, 304519114336255386051309592492886745921342588610, 19037944696024295638921223215273141675851253284790, 1192325081048380267824624452954260532860853011311094, 74797684229545990489671520793606259623594772576379533, 4699536075014230280808674125128023959802087911844954994, 295701741120640025885934424656739912771491386479398352084, 18631474093381670934203978056830259673882340186333628734323, 1175440465916636188160160443982070840187077734566960631187634, 74247644992497768596418930189107080576123014664901202580207925, 4695312158985030208831141301913379695320431574528205064663546850, 297248329514870553155310014963835017526742988266942787730207683502, 18837489616346466695567697733578293058388693314938465793078994987138, 1194957955266156094611582258839410630055661415905004777348028353690839] ---------------------------------------- This ends Fact No., 2, that took, 1.764, seconds to generate. -------------------------------------------- Fact No., 3 Let a(m) be the number of sequences of length, 9 m, in the alphabet, {-2, 7}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [4, 178, 11654, 900239, 76266406, 6853777795, 641688752961, 61916364799849, 6113859987916630, 614832988424140624, 62752222758863566993, 6483650829899569496380, 676834416167597357806799, 71278487569046416052210050, 7563527671079260544924794587, 807900192360879042402313084390, 86798538213532444046946871865294, 9373495517626352310063376929655168, 1016914948923109891140981343513713324, 110779111591445396194938424540449176895, 12112877314225294174303324842177787362289, 1328933202033735326902790559226043731077378, 146249868584252746033461673738170661482070658, 16140275796488178213385486991094726090761476600, 1785870759839218555902390750572943297352973154643, 198072807709932404868404569408651010877962012296432, 22016967921287736258201508352354957271948674625674298, 2452327819943418316786047950673301604189943366462116518, 273669083550055112572634871020670281461765733582541231637, 30594494918430719029211913186681866168714871227722609985138, 3425948220856934093465087173770719377554811647281878444741918, 384231233259139923106138429927197624653074698219408866390672091, 43155634083497301744818019211829021168055316920250040903068772250, 4853752302174675413094717772045913588524516822226461460298045784632, 546611243228940775490029626027564059952288250765869184793172243812450, 61632364667306652590227667250954015071447482196429861698918835589761412, 6957279995145423952752334894776500675866909828302814086200861980564262042, 786219634217570104921868877906838713702807728217106393091956757517061856306, 88940073292894257875286757611342361293804782173705512633366343010986023366410, 1007111554547840666653583657919825918852635697386189771366194007823134600494950\ 0] ---------------------------------------- This ends Fact No., 3, that took, 4.770, seconds to generate. -------------------------------------------- Fact No., 4 Let a(m) be the number of sequences of length, 11 m, in the alphabet, {-2, 9}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [5, 345, 35246, 4255288, 563796161, 79264265868, 11612106079203, 1753402118587333, 270965910076404428, 42648418241303137766, 6813002989827352100145, 1101807202785456951146158, 180034116076502209139781574, 29677341363243548521326632028, 4929368173228370040701922315332, 824194945762220278034556105595141, 138609875913073720834724187387973347, 23431296809668685453757882256965290037, 3979195516110145576081419085787010801404, 678557878739907446871006804106554565878442, 116144014161918661078732815735717072343273230, 19946880766488487470907351142006588389594469272, 3436301356309496414586854590888538252385277822507, 593652121006848612793738697192835416527443259888852, 102824744491815020768125228783213934390784020057554297, 17852543314534175735428393484054390483057958509741557343, 3106428322086405119043633156023563077029024905317298670708, 541641339552696702569666848570732955697512227347586743202931, 94621371707113892778107519475280687150103488298722856263871985, 16559137327192919613520744814317422376543495981948260642152035142, 2902730796801233159762918074469680363384593120736487434864990678269, 509625024375567155720942436707520947187701310436490337878605616407804, 89604168559283702466410297557074305744353739139297894712439943290372973, 15776164466188439735131291116795251206312248779455496931274617826183148293, 2781223628657844615295335846875298032188218610815585100774251150860631883720, 490908000766972298457309488637634765095788699064534216884776946608160941375147, 8674911521868530570986760792702201639160015560107046715614655874366767645568048\ 7, 1534631415594016524380078145175795190552306679830175979013673068718172540051\ 1943926, 2717644329497253484487299158041783211140271110923019962084313349872362\ 199221397781547, 48173470528177124139156706595787166078627347903716882723960613\ 3359942423263579314834429] ---------------------------------------- This ends Fact No., 4, that took, 10.651, seconds to generate. -------------------------------------------- Fact No., 5 Let a(m) be the number of sequences of length, 7 m, in the alphabet, {-3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [5, 227, 15090, 1182187, 101527596, 9247179818, 877362665128, 85783306955099, 8582893111512001, 874542924575207352, 90437361732467946334, 9467275300762187682554, 1001309098267187214993056, 106836493655355495755649544, 11485688815900189437990930096, 1242964338344397490958154292155, 135294163560516394214244049693547, 14802386593672915358103823856915731, 1626958525109139734774927715918218672, 179560107576192929210029836580628234040, 19891067655976720771724498635720956196452, 2210912607893524622765424058253058782769990, 246502063369558281502815364695754079893446736, 27560798966536669949656624489057599283778866010, 3089483559429429698036468167660236517959818560678, 347148306792125191804983135400320697609339102573244, 39093218980356982643240380289607142470275410581244370, 4411391873089411275228337557266673535322459461801972040, 498741562008590845805410864237023381327023182399428530336, 56486593799030289371750854015935460223463413333016415972912, 6408181129487740891550434505216525615777279231396176459666336, 728111983426654738168998556449585224381429824527766455087604347, 82850326490849424222197383994772183835110908256293583591975003439, 9440288052076470059007457988093644496310669273343160983275874085541, 1077052425781243590824286527582049944577137573033125849738247735486394, 123031838485535165133847163782593245862389968217698822540191275047666203, 14070137425809931410188277742815229108075491031559349157506268797495010592, 1610841126594069250357844126575076210575531902387328185989898633227715937520, 184610365184862405796660427151379887092893644406903628167683147822168311629120, 2117804047916389338800824924273207060299816290349501626943158908995863981698450\ 4] ---------------------------------------- This ends Fact No., 5, that took, 12.663, seconds to generate. -------------------------------------------- Fact No., 6 Let a(m) be the number of sequences of length, 8 m, in the alphabet, {-3, 5}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [7, 525, 58040, 7574994, 1084532963, 164734116407, 26070940600055, 4252443527211637, 709846349042619913, 120679177855928146859, 20822762876863605793639, 3637213213067542990001936, 641912742432770594132245835 , 114287840570892852593437353124, 20502971288127330644273350110698, 3702570208952583655019682802997983, 672532254932908563141660483607892405, 122788872081951688284011943771535283927, 22521647274404155444930377651363981888609, 4147941699622771075000003992530419494184133, 766800387134061216482320025909653184030552921, 142232635188109104705825683965953973833993117472, 26463875059802945975078404597473760366912931790522, 4937776281963660271976513336467553127439703168792081, 923705580732071027966091450862926607792520810928009054, 173209614000542548827169215936888121699576157630433856342, 32551326898617478879296315323239017028955263215391910486304, 6129911011171870371923259817902143926140014998339271186854371, 1156554775596959673099699203791556477581134593795757346174981314, 218599371430881496639570094146693699868497254932723833300602282125, 41385848718656958224952269355613775785075930168181874946123308760443, 7847466234668409545697627197553790099993726852985204376621987629956450, 1490185243224103159747678655937642777694272628104244771949841987089507928, 283365227141289757902228931779570608582976046097579173857591062646958144106, 53952789064233561183866731630777941928891255596371438929457807945624570951667, 1028515290642157177019828701881553347490151596264000998361170993811065242839689\ 7, 1962945123277649609850099731161996281706249451162162397100102183495897563359\ 338311, 37504162854126539778436573709374380085007681098732353391157768118629614\ 8582525110371, 7172988500516149325055784137175953787772986763098152718951136901\ 7732179428538299217183, 1373243105388651028290978389691591880902949159157892259\ 7604785096721904650308721106922585] ---------------------------------------- This ends Fact No., 6, that took, 16.045, seconds to generate. -------------------------------------------- Fact No., 7 Let a(m) be the number of sequences of length, 10 m, in the alphabet, {-3, 7}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [12, 2010, 500449, 147412519, 47674321878, 16364395381824, 5854008959789359, 2158646286360525829, 814699584319906115377, 313174547107296851379492, 122190039909499710528063586, 48264334177924170340462530310, 19262229699710958077593902588921, 7755569940093307492453069713342015, 3146456474984388224477947700133171141, 1285008624697722228007548334538017672914 , 527860412546876470188880442742833790623013, 217957960284325667972821721094024857016582120, 90411865398986252298720220909383125342759004204, 37659340032948650719523080741845835169309967550540, 15744901141101837329880568189867867150207241270944644, 6605060746795372911027150835163451280226939176535931068, 2779412599304371252387795007958140372050038731490490267933, 1172883394627705075046553734887580784912846343385734982609139, 496229342570290623664407013217559291741232466939450106430526109, 210449417782724602323713064468814680539537305674889938569611001644, 89448394779985402351123231186901544938583285348067843690463625473463, 38096692134894854429117720156253576293491053162420067611286842114167142, 16256593865170680826668111093165298598160685162887860173865210203389018849, 6949337700809976608605518971917531893522912714239699008562133415356833301195, 2975626039836361102197725366937468200722508305932707435008562045030057563481414 , 12761116087412862154876142706755523675059194411169843900507408009793956550066\ 21574, 548065993581322199177173246973849975812249442423802199167225579452935473\ 408096643674, 23570736957971125428278903917847957150968125320919790211817150273\ 9619684420484354137827, 1015021116515658296465200910060405387229976635176577963\ 48222668750583628919795615732673012, 437629930064549695817625443140132822055995\ 86451860309762238757325639016168562632032224936879, 188903490317754733744286309\ 71253770270120477149479093076387320387782992733264746402246116922753, 816295300\ 2282335069934180956652190818944801588683818543437531906492696419507750201539661\ 778043038, 35310539566797470846978165771216707357963206951776484316645994814833\ 23461276537095808611784981566074, 152893287203944705571060493091726246786893818\ 7623344788423104538643317697420689527873390347661999111573] ---------------------------------------- This ends Fact No., 7, that took, 23.802, seconds to generate. -------------------------------------------- Fact No., 8 Let a(m) be the number of sequences of length, 11 m, in the alphabet, {-3, 8}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [15, 3504, 1220135, 502998985, 227731502703, 109447217699997, 54822481174358407 , 28307972413505251142, 14961013353514988105267, 8053713598352987009717209, 4400469148640852124680601039, 2434155793432667226771857148075, 1360478423228383480823184037649126, 767123247958050727627123046568122709, 435855065703595401584618927244778294102, 249285766614936493695990312011446598428379, 143411293822975276917894372533315078338782599, 82929925578061057401079824561672598836742703442, 48176969480450205216111908529012830942695718926283, 28103687214307998211557746763082198674002790360277309, 16455382465897877271266259167119656558670900753061032810, 9667697993475025207281854833610548411257488085438869640158, 5697419111799915736836231342102599811423590868593827608123426, 3367126696721422843194779442446695445693746231703558222838377905, 1995110949570137201945825926674383769903324929439617712004889384559, 1184984820770737716645798895766818385570139863997056696087707095033169, 705372401395723870400568779661779834165701893737215227489783516935203112, 420740575186479464188144698239247270600901235455365118825437792747824451697, 251442232923680333027265199135209984816324563881095275026606424920258910384649, 1505337631266913575496591136416708900978110761229109776231630643220248156229847\ 64, 902715604174750059889350950791978341159249955787300489538336662214693750496\ 96582006, 542180126971371416628714560172628501683451304209627912037493701120042\ 11178596947248120, 326114677169680185076648095637813877306872867810506000617403\ 57403244926551261519822783848, 196423480793720237408590263197711420009983432013\ 41955484075827406589565728122318213474951429, 118461779547024178162305805031137\ 09623468347961646256047023948636644201075489198540564288634586, 715308343563189\ 9543425776478817882359707444214018716264072869503388225119839808070213627814622\ 977, 43242372050921091597166657844270166196553713284000805183653750955679629883\ 92910751155739969753292382, 261697886719263121196088494619132088976940046995481\ 3377771953872815986683538223723445498535765040539473, 1585407089818730899607219\ 0940750250381686664439899371419150940808495242655904892558748903685569621493140\ 39, 961409707090307397897045220151724590786428420315068543704939123754590354399\ 280215572587583777211696978428525] ---------------------------------------- This ends Fact No., 8, that took, 34.945, seconds to generate. -------------------------------------------- Fact No., 9 Let a(m) be the number of sequences of length, 9 m, in the alphabet, {-4, 5}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [14, 2529, 678338, 215205020, 74954297074, 27706901431307, 10673513460281995, 4238336602544646985, 1722532440479713492905, 713032912920984966665914, 299578160956737689390169226, 127423760336300467698086830007, 54761998197408482500643162670932, 23742924018375726263229745697113234, 10372639160995862858028814581769132028, 4561630228928173937942703373111379977087, 2017804841330203670735598373154752656772860, 897177656970083321768073667118946270564224913, 400752711842105711698489590738780696157143645906, 179749957957898273181768638889177302565608486108515, 80924653758099087193611642848590771414665085220443850, 36556309544907124424830117175556930151453007265988699667, 16564677903334456428003716709807760335580928291214833115606, 7527124558532594346536646943982231197215225418918819481554586, 3429263552961285648003166539047407716193143507178037104769111464, 1566066468328414529702748500989967536486318127577036893982829506888, 716768561186324096677810512889381831577143689734651074373928490318363, 328728612288833087716158981552700126236668367424974904178777273868860162, 151050981487842905575015897230290843691016552943563178561724675552959853422, 69531419296511660079027442856624450723039093889934179554973825253086323536882, 3205971393191855596725990403088140800259709812310910041780490282408652646087900\ 5, 1480517361897073351642795377619244130841205806496674146651805588046826104876\ 9846236, 6847013721977610481239691270069788849853840429316336479624588553495086\ 117690434114290, 31709175533262555129040382386449438541638428763765722409295091\ 30571341826036257044411405, 147038230014790423335740034354746906528402058724794\ 6086664522803937136381646757868235844855, 6826617387448108436059268355784321786\ 07987418385319476103665036570591784261510284215459193247, 317308603008005127730\ 580446050869894354380179457764765538450482132798403090921806139978642449929, 14\ 7649657303911809058830775167716604963495785141063858531825532665119408147417588\ 832638856283145837, 68775340703689757787558754297021032852518891703833467442240\ 751136159865600117138791257011101003259150, 32067125645254741510636930494465813\ 565258882706430558299867381661203800505694950425574308496770710077485] ---------------------------------------- This ends Fact No., 9, that took, 41.655, seconds to generate. -------------------------------------------- Fact No., 10 Let a(m) be the number of sequences of length, 11 m, in the alphabet, {-4, 7}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [30, 14985, 11192590, 9905559856, 9630973835211, 9941697669557915, 10697085912467428725, 11865690376622216691749, 13472258390481755448883694, 15580562283781219754111028235, 18289528657339881354898275735265, 21735782261207943742126842927563367, 26100399582081198684530198941275298527, 31619379472450558501949640904593148843583, 38598131733894049143869189347797241848293666, 47430786775450230884905495648227743062367559125, 58625567828617800798962606041777553957587169569324, 72837944502462959870697859797241240694426584635229006, 90913866280501457946831100036038477265964933959060551467, 113946112115065166281740931243418573887824126508028593439960, 143347748141869937171437369598917952372899163847120594715163536, 180947934020785552917144095852834028983043451373540413110226956320, 229116954808485179552184410563550173809248363586511814228565593142848, 290929503921578165316808771281645271992156067791538589561975135654642695, 370378066970518506323376203684694470553279592880309335078771814457536705350, 472651971161377767815926195542511932302242511045092790982445003011360002229548, 6045025543362827837324205551338007299985138036603389775706805029483490646147543\ 79, 774721346140888174934420442428729770927074012824650592458616096171659014900\ 030506352, 99476663601611886011175602923110164582726202508819758623401760609553\ 3978110512768975361, 1279584982444078061576384351368012037122782964953556019861\ 163182868966683378900532327558486, 16486889592663058512059836779160599577440763\ 87166472679516309351908535870707304877850207033994, 212757188058192120573075960\ 8532831933731421240335832874166211315003638810329644799472011965774577, 2749565\ 9071507011600238616097341825887288569400509264322601096314236590070712950821986\ 43462357901178, 355828381424261124296581696501602731179024120454390009233552797\ 5068483094388882628379999827210300269998, 4610829440475328930638346789502736047\ 013306394849605804298558788316930286583295386733785109830575127178961, 59820209\ 4391243084233440734681737160931060286967020243944892265883352469755323490048360\ 7437692357954494058621, 7769949106862360701060102880195905334426608681723881099\ 753268907799669160724168662546076836547206923332203018516, 10103296202972341201\ 0007901981425201857020054157075956073267299354648013889337273196486302557575521\ 04306969796376036, 131509775168956140524609654732056112810163680488928597959721\ 83766518123048282250429751954955961803206487759283468471168, 171348482874110715\ 0303542971945522990954629808785630061539449681353066285890136295725818323550502\ 1443725917385292372481507] ---------------------------------------- This ends Fact No., 10, that took, 56.719, seconds to generate. -------------------------------------------- Fact No., 11 Let a(m) be the number of sequences of length, 13 m, in the alphabet, {-4, 9}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [55, 61600, 103609399, 206703946565, 453232991131761, 1055316971192365243, 2561590923717235242675, 6410485049302092081392454, 16421503639147163921590284676, 42849423342281218735447266483263, 113491772187317878291880313352241193, 304330262212031300714767379651437328717, 824576896245683919380712954926256719009442, 2254012437694357679866497988406980415928695382, 6208576839762250714817727196319856252114923502897, 17215198830413977210317119752369062959887218419062569, 48013933605349586572662705480209459018689541874355412796, 134607283838656243311067483386393117624561322422564332967655, 379117179464470215411612031038458840892337912937344797145599948, 1072202557914191371112256428295352449917884307005801710506218762977, 3043711642445247086253114993329628033078768806333298350084503035078117, 8669672574021807225137870206170845980952766510311793941324634807475776786, 24771010137313568285889138986212905621892459655232398861519711067054288924519, 7097616641088683473845073503644181656314485725808324216382034349938176234562901\ 0, 2038961728919486241120732646049384592881100880871196701156109726285743694132\ 39121914, 587144502408474915749404727542979131042786887347224147101073573208881\ 664692853952442200, 16945015617779118658440117865680050540843314493842289173710\ 79423032396459702110924841374128, 490038492398743000333461343414063734871537813\ 4385309874457194077738474015330538676385603099259, 1419865477155602578901108502\ 0401616226884493068376003067520673218031300935074666564100027393026599, 4121325\ 0487749084207416943107258612839995109927949014668028309369052808733339339808667\ 740602558410443, 11982531971482723192323926707505513942577034284773092146141734\ 3813187797106907538668864258158298194472018, 3489286194597683994810101750335697\ 19412755261489356639335405304148221089034915584552194699659971648442596542, 101\ 7558267447437878935209159791730714683803321200243880384802756798294292810612442\ 141873981575570377638754976619, 29715211899170036286450275239234621277234169891\ 54352783875409040640958983622423246357284012358258281807026221301664, 868881725\ 6233197159896179574232782658605825974721976018427037193274243481139556806213558\ 865498672188400477665404468370, 25437424519461235627439510710722445428491350232\ 189578226011616581492608331246704834248590261934019964398021118409439908414, 74\ 5567914129322889744944643335903618601993841777936933005358445498598986553426218\ 49906628741778609713928774595412283565616217, 218764150331731178621529388749452\ 1812789885006267011139599120563777649372830211525493827169769186455756217681415\ 77657847916262441, 642562426927579466813377392635241428711241188283824717224431\ 766416559796677314813606941165708003109466224406925020100311429548358113, 18892\ 1747744893774886281369264725029587709816861853448252938358879626028031778813091\ 6923334800562109916106043115720849023533548870894132] ---------------------------------------- This ends Fact No., 11, that took, 84.079, seconds to generate. -------------------------------------------- Fact No., 12 Let a(m) be the number of sequences of length, 11 m, in the alphabet, {-5, 6}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [42, 30275, 32675894, 41801669777, 58757395707430, 87692166325858602, 136424052628767703708, 218803633512952794993022, 359207350359845275021268732, 600670500086196968953098493461, 1019548167065036208950503773416034, 1752004991899521391756122592044282946, 3042034836797842895970861639014826559726 , 5328794951781120970145587503377874801450823, 9405934110291932630862957007572132670419945104, 16713101444911927790135600872984714010286956998516, 29870724321187795475481153878471983310142040026880262, 53663512180991572550190646088070609863531258865434988521, 96853404214631891450758158266402354962407823477474965710806, 175528710156446338907262214568649539295226404764268637894542861, 319303642082980165936522676301342646754009296188477216790134332946, 582815872213572064485458397632325198335694859452568698638472324021919, 1067087567961470357617947611615958987776270111365984028027267828060898126, 1959277677354740929146413956866455507809547782396809576178485755763743314648, 3606775767454797616122101669568565767789066955724698129088250377387176276361306 , 66555086102267974685053005247852426460055467128132571270073153883577921900373\ 00440, 123084685291250061011064067384989124323367636706445702362641334817470424\ 76450023699668, 228096011664044185878114838427029726499475141852389405881368997\ 38583263710927675034234990, 423506263213732814048874432443265452345640403289303\ 10191076778547584577859644715491962812846, 787724529353649785142011772059678201\ 90788178388356831550634851949746319024438346349248331301589, 146761015817534581\ 9747937917208458505086810099550562665495981089800621604516913169789143232421201\ 02, 273856522598535947400207888614505916066710996315742449052567685700922072708\ 396304536356183533538942681, 51176427066489831245647439545887242467507599579869\ 1135462172900099168436608067289912822109845687966590140, 9576647353732487752922\ 0452934949745163729056943916142240674303677763132070889164088805704537881922788\ 9200088, 1794399518319411595109467441050090048825386541452868437672587819679783\ 606862964402793762105777567235322815818982, 33663190727165960904693304610596949\ 6660505089891099471821079229818725401297764517392934121301279936208625279977348\ 4, 6322558603245674259891316086594130349296796048271252423596631697017111622499\ 286462336287616359115950624616478622938718, 11887901235326279626752754044682238\ 3430133918995583569166797591268115689479644097801074558402384438897895778308226\ 23993294, 223752356812366866514861169867052329165446115799095809661807459721728\ 24821659514012635011969784752643581203382565381719432058, 421558052214313228701\ 4787240518629563439377945379086680226254869161089236201582197287896601710855101\ 7543033140007963693870602826] ---------------------------------------- This ends Fact No., 12, that took, 102.521, seconds to generate. -------------------------------------------- Fact No., 13 Let a(m) be the number of sequences of length, 12 m, in the alphabet, {-5, 7}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [66, 83897, 160105330, 362385581976, 901469000985287, 2381308530355660631, 6557587116496655764179, 18617682359843841541191233, 54106242347789971436724365832, 160169083581123159692622245179870, 481279111642151405816206502961771235, 1464119286844987650631501908617054682952, 4500502319411583234691603104731386779127533, 13956756578133924098721324321280989682928600568, 43613256546412983544081319441944457412473522018581, 137194490756831982144833573184990989197896294471821527, 434100233833183936634421591923866730125622766869908259425, 1380668252911888725429612367843822308854621106525731959897271, 4411560507262902230999018299281852922501024479836657178770421632, 14154468686935428457682552283899221518033479944461867142509732185310, 45584601425926500460247553036908783508503593480119391152376458248387403, 147304239508773993575103324660094803877803482088775632425604560704342921796, 477478846890087803921245370771129558611403600407519188376575542554656648447301, 1552103952073855375325868062449425175639927091008783856644734531453311089531259\ 038, 50584232911681792617124687986938469713521211025531835847504408109163193398\ 32893431824, 165252829477256224417765316405089774120165286130042494717227685076\ 29692293725075410305671, 541057816087061926243920498915210763840441279860135022\ 85910563047912410453509838573561069667, 177512802292273192988476368815981923698\ 908905115722825443028263391176075957220230285607711193607, 58350483114673565434\ 4649510131908926327364587131213339847224612946733498864875629183738786503376639 , 19214623801146455783078479084524987256232093812964720310091096397933147184113\ 18092959430936101969824129, 633784137219650313688343277171980302901307020904869\ 8659123503274089174458898374919982610900619919999542101, 2093760989408847838509\ 7671742744653662361516690642435438427298607379587579151892488136543670531314159\ 633305274, 69270374264489412737194489731914782483191067108653392450806827127612\ 301677452293997597653289049977380433376412331, 22949054996076540317095795835077\ 8352582968052148380759419893047736573101594142866197315004896748355133869228955\ 361385, 76127989458502531976487465879398641369052589997425109734615622648008462\ 1329339166736346272027276706030543116524552690456, 2528451053934156191573842168\ 7921731253955265993648193775893701203164158945257656811798629544435605621993303\ 07992153293080730, 840748949611885019742777877337481392458613437530332255537415\ 0509261949113528716799468924926999262976274838941344123734122331114, 2798678320\ 4171730847282290850854525778257542257169155336962583271893287734683828178309737\ 837927291174276613249885619364367976483108, 93258807513690134856544653303288717\ 5777469451470675930601458855530000028789174112458246607989651206299661490218207\ 56408015831337366349, 311067173174111175818608231214010615115589260859962470463\ 2402263688905760331623105581003902527824574624835713620476593152698518411853543\ 76] ---------------------------------------- This ends Fact No., 13, that took, 127.431, seconds to generate. -------------------------------------------- Fact No., 14 Let a(m) be the number of sequences of length, 13 m, in the alphabet, {-5, 8}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [99, 209198, 665024109, 2508666567044, 10402794487299180, 45812615376305774471, 210333338412266065179162, 995633490114825549146199962, 4824376491142825908775913224967, 23812199080824762524110671812670013, 119302609600170515809386854882688807503, 605153212891352095572925555460830998620365, 3101621287613694274675333304166583588165354377, 16038121228779473509691413799548116187734910273833, 83566182067440671475991826510655114509170410166132863, 438322004386974667063569827743884201729088516581679522042, 2312556820001764119796229240039242694913110323692476577205151, 12264180968243874324257777850626444083099819487989308081803596385, 65341505380113354470965376646703041118094729884667863051152015803921, 349573808988319278719270630362976176249012189140058100231990875531475560, 1877208557225538662879711605672125723230802895461823270209184716022852916449, 1011484430127195937740507038589781030470615319029323970457373662315300955656300\ 4, 5466988456727844648682030949692620966522104141241058276906706852640661768073\ 4067760, 2963229600697006720463080030714160079387577958646674288656836103940926\ 88872841952271745, 161031377761774778824513376853303719807633626518903018400026\ 2786086617600929824032495359256, 8771935055000858672956807528024093831282196246\ 417061872736997267653780111012075032893043952278, 47889646073846143293169129815\ 664261426480522774850287546669694917837082666423099112907593346483595, 26198675\ 5250649989565642883636948363642952931378282553455319401371054977783030721559753\ 049105995022283, 14359722414957448446839237421681329605488719244274023620754266\ 54140899516812273347264249821404912618002585, 788470796807966844129569193858725\ 6652091652354350778436571444015715644287758046570104357086602192043558398703, 4336580532041167494423937587974195814297124449213351882157372628721277739006770\ 8370540738381969210772709819922905, 2388832212436506105320524595663748553563727\ 17642550540193386321818439673283656939119880437689231058194415443606294049, 131\ 7827973772818147702617156804596201689292393992905163684222525237764306757567641\ 376309999508268984479482331717908562606, 72799564180508707440800772899800611443\ 0521761069784792040414193513375879634268248366393841618910159004113848589122898\ 6741427, 4026810987914640164492061315691737627062795761188869961759436909940704\ 4157107858675229525815252990955330400937762379166913123511, 2230100651827658066\ 1146779526400799521851092194504298338961756867746462067280157030016463046620838\ 4417497872195087334934775419633528, 1236486512299625552619710415846379612192961\ 8427185894030731927168835052875720809786019732693353941214758823078265320534148\ 17850506875992, 686324130832239775034941706442787377215735501932859720154868004\ 1851780512464437915609495768089535098939938160465627579093147364966972580034, 3813462406470470423488989985553243398576360942768192542192495042821776440135564\ 9536438765237406403076486821521070339723260835143002922338882860, 2120983084319\ 3270941101749257193599053784789231455872186936995302897460929097733340976632254\ 2973088568213134881321414312251232852062394418665361785] ---------------------------------------- This ends Fact No., 14, that took, 163.417, seconds to generate. -------------------------------------------- Fact No., 15 Let a(m) be the number of sequences of length, 14 m, in the alphabet, {-5, 9}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [143, 478907, 2416852480, 14479410843183, 95372340392047316, 667200352642839745369, 4866284734970204704928549, 36594786835385709997013808787 , 281708064983087336237083879061640, 2209029761751854803146003586548077276, 17583286465732143216602423151971170037226, 141698921777543924527564040472535232273178825, 1153834195986986450963924286604587809220194654852, 9479019156700412966788296953097172845883982504469494, 78468835856833395487043946907755506298696101633573837702, 653909810995783144634500622679940290491987082680818273224940, 5481203214800591345630358175239455461617495309987668570866319446, 46182960532338037109293046184421996279066304253044575484568357630457, 390924392612766778308310877117828882605630276980180432383750765618525460, 3322794778573132246891144473961948792989137217042367643441555222356590845628, 2834905955622873993718865554543288289819915681474071169930977726187106619025996\ 5, 2426874190137728299719590468429336648307329620319624933254681730527363416747\ 98725583, 208400387826615285743290838732589285847488725528215917920206875151574\ 2802659877623514775, 1794643962893167708198711840380903797942136714285397876379\ 7239218933587919724965131843796089, 1549481621330241117539678233749627038629542\ 39761491363983101721197052203443150832162362444415438, 134101768463046733929871\ 8051510440599597384398030905155014227408189352976710704494231113938052622535, 1163172507040112859143632445588699708984601723921619137038842386850942658000933\ 3101415673288878167978555, 1010986702382462928933294436094747077632405594851624\ 62951270384941846579759535966711506456747859427323644104, 880391699743980169676\ 5583619643735301577389287394385787619516626346881009047683653222304963656643596\ 71279675369, 768032252575624745829427819067765824405144032392262808122968450484\ 3443945000634932807180946232122307355104800188699, 6711280016439846848250671189\ 6984055500290660750612327530627409239127167786169495085467014871109169715970789\ 691891324490, 58736467244105351653159964227852011274090454409650544249454011310\ 3958059559269882732777817848896727532441314151655288023909, 5148079565824807754\ 8622904140588056399311784391025315920046897022404251059339490922306408416567005\ 78626403813901313202153440427, 451834898367938701331369028332751279016102600733\ 2905061953809231110163387820607871539307033877622936136594675143043106679617841\ 6935, 3970788925778369497119074918253984403596921300408228786861839323509063661\ 47754302991285196723998737823140823967191664261034806962246093, 349385340777257\ 8039315893551114502223584780794523443521013980371688877648941544290080955465806\ 491144290514630014489541851558449340898981516, 30777570927343372978918036699173\ 1964713616544976885891707145411156001358322028604866690964495558814787502922820\ 44408385612663048123785048731258, 271418250877140669883773858892167396875438175\ 0875942940281170839618539863419477432577541311492918262265408841422396100038716\ 62829203173004079534846, 239604042890958621726215177604750534351379417915546922\ 4156150233569418593052294511420177578054194096103353924699058171018454598477251\ 498203720521900809, 21172728948932461072548249477571102067232840759613841030758\ 5382491024463578789657260106947741389155211944666490175307736661618540059998266\ 70865758469579203] ---------------------------------------- This ends Fact No., 15, that took, 210.466, seconds to generate. -------------------------------------------- Fact No., 16 Let a(m) be the number of sequences of length, 13 m, in the alphabet, {-6, 7}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [132, 380162, 1648367138, 8483009184519, 47993415059472400, 288375592461146982729, 1806479813212412643936336, 11667618233974493344600972921 , 77141079627228888933458478398850, 519527684721737559256633776194067055, 3551615050046346656609154959590709316458, 24581569251799901658752169800070786335899939, 171910557387727946639038296988838796008122325900, 1212935578919447844373532859252894021151542403011580, 8623548772094351155907519972754678646969610619428459090, 61719240130503928834196210978828504197349049609065341987776, 444316098784448969527750015383710618916325390498034714844630380, 3215221880256941142248235921190136339863235070970192506188871994501, 23374086704062515417725850624526028918327288761790377095163938419363517, 170630954351678834104503006190102121386568778888135612844069088404524230974, 12\ 50273814272158206206415662814375490702322822468928174615641033478684691180986, 9192332236618511491922740646193325300098710327815245875165487713998669637344005\ 069, 67793610238139164609129808639771461806875010918763452010118093435249001679\ 092264478615, 50139520151153946265127432145072055404721034933746188811579379826\ 0969257254865059104050516, 3717917880228504593105103773180015863464241455875988\ 632750668747293358959398134002095695520085, 27634985346414337623305689763868420\ 254121724448057576150832332488520322898398324939932953505699074, 20586386527202\ 0256705723530034709125950271354165996062821712250436320539545074325427871558471\ 584641103, 15367124182295156169970754215530521557494876385822447802953071526666\ 07980253644166349271088811824392104616, 114930202796569553196678338829494043354\ 45582087623629839403297129693501848639822385186100670253899361710986537, 861090\ 2846867165185142108925928885188199659837443460312536476986469919845258745317692\ 3369134237848931893429257071, 6462276220387588563074311510117986069927842764643\ 43024775502565943193620917174098054085620170434201739583650034912829, 485734378\ 7003152259591108339491604311761469620781400086302667417819194556597542372658546\ 011556746892651995697864447419135, 36563442238541880414157686233481123021139592\ 0028947654933528856905981483596795222844991780985930407945009456677375072403720\ 98, 275608334985037721767062481747642545482148041220706062625323516570327836253\ 662073929570504147644860053993716330231904452032721182, 20801759956460820822636\ 3648108161204323621366928582561364145406944885093769587899281535977672760857352\ 1870585275848271915540370194424, 1571949593715481937631190511895189620913349456\ 1515120376169025482695842328030622950567032551703567617743244040964650702322521\ 107078709028, 11892657029681513966949029415618779661110573889742190044882674463\ 3079327443374012124039738439175261851982385276643347338997688461079283874607, 9007290726828824390156939543359202245698221021075790980948017113893238501643252\ 18952081016023020590134104104928886241116864393057250043827880375, 682904256587\ 1555001288480103778610225224810714824167191911753319245806781725986967617665223\ 200858754126224115244651635355990402556786871613887463703, 51826636971614603968\ 4183453340784513119554155338193655679155528477055634032071139090947134808635252\ 43867023430755796865033324438516051039032251003440278] ---------------------------------------- This ends Fact No., 16, that took, 249.362, seconds to generate. -------------------------------------------- Fact No., 17 Let a(m) be the number of sequences of length, 15 m, in the alphabet, {-7, 8}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [429, 4939443, 85951719854, 1776837502896323, 40395836104890721062, 975550902221726834218742, 24564396203872838497856371252, 637771018023375594167312568173747, 16951046179987736828466213086147446012, 458945089856316870922716650411834102804278, 12613296473941904086786679032163924253005494648, 350970065567425413492782395663636565182075947697862, 9867933422730857306997293526949017221749863283934313392, 279916726938354140025781748497359516609206296674038444727268, 8001068737496649437605678367429823283674454077619665860583024976, 230226958530286794430571786955631295177508207802072111681097540150163, 6663517761446602665586961431213606227359876817198269559484002035144040449, 193865379601291348697730111601527070435290997812386927918035667697372581955392, 5666349500832899958544421604803973433719818631949006273031606836096804654700596\ 314, 16630575549273036161729515926461117297813492186207466921076654991094933581\ 2343297920902, 4899327920618189357133623774861765511709982977088511392661541224\ 764845695480697244845148262, 14482382103388285686754302959194995207119636990915\ 4940487579507465179890511757352554710264062872, 4294245199602266240890889464488\ 522991779731117579202879204375208998466596994701744683323207956961308, 12769173\ 8487781890287315738000584601875334505215194828206705937076186059773017939997970\ 543997291887833190, 38068664365616302293687126483124552614275376430545868321796\ 14169161255553984403780531164137832929869878901536, 113766107491686350837432306\ 4481116861876441844664691209525851453525895203478884617827199851827370174313544\ 19441072, 340737649760490301847583789856143518060435699490765024325673887601159\ 7751528227961298876777261303069303516308867315040, 1022632080030251649047753173\ 8489631596954469464439737851139038722879216551101941445368534107037119854516995\ 6306582591220548, 3075021928442932136090504731824547044663751205296246498433409\ 435892015888268503564681879383681627382269506532991708463296040168, 92629718731\ 2095478585806011338702709529920824364421161803555827455969838905085535411046999\ 62943785810673671502400978431124615958032, 279495698185301145268116078712042819\ 4964133635864034862728371988986662357300593868709983206034747791405691509363791\ 176684331159686142432, 84464986701203806607464936906084562032294051100164230176\ 7248569883190908414031015752000609888990023696952969011876730884261060067852326\ 20883, 255630987644295159453678363915753021538480217744613408794546415553302777\ 9429929683687272537509990558950537018533095936176227442464930781039788571, 7747\ 2500665771571426642353245750586122841001190111626232539987322785252587150200885\ 285604342495063523855677481471292749242164569186283240270143463171, 23509551658\ 5513714215015077302885947925204623331905747964185903771980994846168268715034008\ 9145028750334030952630693321544282681758849313688835115032283552, 7142859986835\ 5216524645435796783804398539965999205300923394699629415347306892601172619383510\ 958087159638112736930079322373453811847029810083620538716942983600, 21727105289\ 5873416776100269516960601238014432012842359887775121036460924436308013112965094\ 6782069231311289657030856929697058556117298334974901317612368333951780910, 6616\ 1720518623532769099566596440313469677782154687614052143972591661889488368238598\ 5478176091078908218936891182605650390090417844244600250725797372889584017468138\ 14002, 201679721593423092985429160564938981159325330262989538426001181013155623\ 5467726368802318464352725672139987531046456186946661619847694816088601981028551\ 234497294494338251288, 61538286750321946645866633629450878264909465159846781682\ 4277219734480394951834921342647418023990045802465933042943256133895974938477661\ 31607869565759511590061617926840427496166] ---------------------------------------- This ends Fact No., 17, that took, 325.852, seconds to generate. -------------------------------------------- Fact No., 18 Let a(m) be the number of sequences of length, 16 m, in the alphabet, {-7, 9}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [715, 14987975, 475474544886, 17925472935018957, 743302611311730852648, 32742572266413420240262148, 1503898210475254241401678750632, 71225641607737412000274647916943710, 3453296067639706949514330525606137957267, 170556498193011608092049365275084731665619703, 8550857341259711102913455467405274222302562736226, 434037348619516824083584652420450557123802742165516683, 22261841593359849898686483208636518183183233006000319901800, 1151975014416050145836254442054241945237956389878101159160017624, 60067937748467073152742759347453950610917005696383792674075868401015, 3153060963452612249610746243263642251021377987566292881069223160865343599, 166479978531350264083085806657437068207555428821996231138937643672488526919253, 8835713558969881479623590347355418921857609613708286539769351677561051033201602\ 333, 47111637130993968644067934258062690914227412209668063279520571010680696791\ 6128617909540, 2522412189877662889917242834435164616456373073956201987808721767\ 9202046347905172866372307779, 1355593965486085291687306853025486012849658508206\ 441338315090921395380640115563238422138488245396, 73100072654833605351744037707\ 581896314271137903807255223449414962078486975646931306558063286733598806, 39541\ 1809543738789628768130051201872817359931630630414710159739847595656053366267179\ 8462942992103491724711, 2144918777227243568737030494690521069902584700489300287\ 27033264728062430622855297009961426306579511715707573628, 116654414337990253992\ 0339796371813080055728231089942156981417771218085052791699375763975734986915282\ 2297627829549652, 6359634085452890585158592519305539158764571629828451726988915\ 42692122332977342629488280064645896239815480905421554678042, 347476330885637346\ 1383483785953186647608370667015105739262795032691507676259363811173173712056342\ 5695218606555231616811968764, 1902438178380274924257429090565497603187184076736\ 4059996848252559227165396480562474213415463315418379234977924439125089955735102\ 18, 104357950269771245840638421459263332054206855546275199490573530321564449534\ 392259850823872453156632270465551696930937319166783545590451, 57347433007074663\ 5242195866220127357538916343059912564152251461473871949735443970297098126864427\ 0556121632369623676745858339433882498288562, 3156641036249509239812157974251144\ 2611554653937661506692482093568971532415652198359501576581894978774675461950452\ 0542726195286250576998765600512, 1740256755474633779699832568682439245719760122\ 0668617808887258780145903296270368587506677182684707176147877712005488304295096\ 899175940587626287069910, 96080829991664087625106233093260958791445563390298480\ 0672876725356186630967036593746609563290577780718137957948493921269324632526808\ 880609386609710777445, 53119936758103077820009722139529181656824667872359158458\ 3380988667864381164509638084157261355847700108365475875002071098676707543535235\ 10272474829957512028446, 294063505449764096435162195788573131448915634043582325\ 6622215326055532806759145434294166325297134245360621524489980739290925467607465\ 566661798253907417453967710630, 16298806471613174886463167665173370459372881603\ 5077241957198186570416644753458555149035079270483613509288526923692657363692613\ 741269467416855930081662899191351413357291, 90442484898951038559194634326921377\ 6122810664022354644072466132329887395214208280257207093371898897800071271393916\ 0519124999123292140790724540017458360675295099850734902395, 5024171109157142218\ 1618313554274339494308942099952022957100166891368336687757781080066349529655580\ 2930768772240403924089077503313444497153082767329383992374818453162244098779429 , 27938747785945170745614299334220826219588373288359963382810822228921171122446\ 8817170912929773513041000589123742925502651313022070673519890864939925463462739\ 44318771033993455782206667, 155516771417612000956054587053718986420303859745550\ 6682675571622178659544982875298129467926607501480819446843606535278840511344568\ 117364319927263305167701763573687883564709115166650195097] ---------------------------------------- This ends Fact No., 18, that took, 427.707, seconds to generate. -------------------------------------------- Fact No., 19 Let a(m) be the number of sequences of length, 17 m, in the alphabet, {-8, 9}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [1430, 65844845, 4594985616250, 381204353170196446, 34789095953202726636316, 3372952277607889298001521025, 340998603450621439510792428353298, 35548211785115597587828534964319551748, 3793755978957247209377113351598444396307206, 412442012632195940454444859341825989923232896892, 45516290567013158925520793551476322377269489243798568, 5085685678556237828317436100980927343964881710376797660544, 574183543975890744664925739503438393671577884670231812501398018, 65403583475074653112825474725899981448039763275327844055058844017555, 7507084578139431831869691846662430286441912844510835219302589254217475359, 867425308101807654673481921398705496576433671380958696144184999357898691406897, 1008169874655501920624454081192278820297655525332090358478178385553600720979546\ 46839, 117783925351564582777442732461584001115062837316405735393652299775341083\ 29570087748139654, 138243868337273739131625347424941601628552270908641342121024\ 7906370136289751561237588380460830, 1629324280263676829933180925034025378135014\ 87166140649209025938179609654014587037863945689875951897, 192750416134177555802\ 9419418741059511331962957955113667132678908574475811898712604804139534892575676\ 6397, 2288010981125704214727853836450573649464514768286690029489327025890171564\ 106762216911846804478887327995383110, 27243622588596854856775543713473218246648\ 5284732072622502018763405862861718799902515338369327460701772323256213094, 3253\ 1289468563858257268382554487621122779576788482322871527172242916166210923100966\ 373202215716782672911158189375316748, 38946376889290953829221068528650896228830\ 7735067424148961199392308935039442963880655658101757033334827174093729127886429\ 4252, 4673838393304067201038929641944995353832173541816730493947518312125156537\ 18685066413691577500073026558868921285556805354500669892, 562137639592886360343\ 5846519389796862903672404491216994413751450661730230534570582392983840701882535\ 9926754207034130434899233149156686, 6774917986914391432743549339199399909224510\ 1356783436140579514862596882182126872684508823861340109233447714109439699785910\ 62374312648222543, 818078987205928181082637633511586245634153900876745925497262\ 1859034421607893374782239704266431501712692598115580011404645098203722439103498\ 71526, 989600781117074143262734654527292266480196173373591827056001850243677688\ 65067721302486556323456782579025329718113619943364515491268331135396939814706, 1199078792274091461153730349282039492594488026737687941289767736088132332907274\ 6064949489445839964309078212240181271538059189974655196132680709854517009988, 1455165685241958905162003745647630150063310763105329294028380547445258200690310\ 8880914247640539876924740665395896414455257884732980898648862512718880349766004\ 67, 176853052707156309484491104453155041725679586560911442012032217524494227485\ 7499488928915723421889495014901664143425530059709830105495574548222755884484793\ 26350237468, 215233643661726352506893179214127190351903960203920798599782774298\ 7612112413062046855561279688924493946998938520171351244610481724710344536067237\ 0499408625907657673748145, 2622832884737838889917454140708325131253884814688877\ 5748614950751175366748592282555957934227506714297673422760547005344612318352674\ 16754976407129722681908777952289518050279098, 320009046000112396884777411034464\ 7148046640026044205736242187587017642454382857040590238944424947538727521564261\ 74118460661108285153323587395656792367182753419798117053625209840070, 390891272\ 3561985985324969714905481848175677691305772863368540161490712069539514884247939\ 8334901885484264209302640542666589452058519752292440319016463756260777333936589\ 819074820095611406, 47799682654688628585743774250618521027027500542224720586746\ 4665536232975374822965117267933025234027683119233992043669959918126961030737844\ 7712839879169296477057210471318307844719913240646151, 5851196659807670870023978\ 6511156334197369887689140678853312874196365624741738046963334917716885907392984\ 9408551577217319621930758161654506007100296966978460932886887365397023119755002\ 619335108574, 71695552189852202329211848205379313367076548245637662472726120228\ 5466915144828681999416161981613725204491119175979936179197686942615638728517525\ 00461134327359408566996651954169642059005671963623912108] ---------------------------------------- This ends Fact No., 19, that took, 566.758, seconds to generate. ------------------------------------- This ends this book that took, 566.702, seconds to generate.