Generating Functions for Enumerating the Number of Spanning Trees (and Sum o\ f the Leaves) in Friendship graphs where n people live in a CIRULAR stre\ et, and every one is friends with all the neighbors distance at most r as well as their asymptotics and the BZ constant for r from 2 to, 5 By Shalosh B. Ekhad Theorem number, 1 Part I: Let a(n) be the number of spanning trees of the graph, G[n, 2], whose vertice\ s are 1, ...,n, arranged in a circle and there is an edge between two v\ ertices iff their distane is <=, 2, Then infinity ----- 5 4 3 2 \ n 36 x - 132 x - 46 x + 353 x + 116 x - 125 ) a[n + 2] x = - --------------------------------------------- / 2 2 2 ----- (x + 1) (x - 3 x + 1) n = 0 and in Maple notation -(36*x^5-132*x^4-46*x^3+353*x^2+116*x-125)/(x+1)^2/(x^2-3*x+1)^2 The asympotics, in decimals, is 24.5983738762488433302504551780220192949234009778633914834899348497578650910079\ 2694677927924608330251233229496882925580927339297248445890624972498066843398989\ 8767279952555335411962021549274519211192946687427296435817518828658277354431475\ 3789650348892594497588196473457134237488647674990022417080426495*2.618033988749\ 8948482045868343656381177203091798057628621354486227052604628189024497072072041\ 8939113748475408807538689175212663386222353693179318006076672635443338908659593\ 9582905638322661319928290267880675208766892501711696207032221043216269548626296\ 31361443814975870122034080588795445474924618569536^n*(1.+n) Part II: Let b(n) be the sum of the number of leaves in all spanning trees of the abo\ ve-mentioned graph, G[n, 2], Then infinity ----- \ n ) b[n + 2] x = / ----- n = 0 7 6 5 4 3 2 8 (10 x - 67 x + 109 x + 99 x - 282 x - 30 x + 145 x - 40) - ---------------------------------------------------------------- 2 2 3 (x + 1) (x - 3 x + 1) and in Maple notation -8/(x+1)^2*(10*x^7-67*x^6+109*x^5+99*x^4-282*x^3-30*x^2+145*x-40)/(x^2-3*x+1)^3 The asympotics, in decimals, is 12.8398757751993943256584201659460755580689808756811940859001840667823002658368\ 7811031351349613089295191218354731422849649224941104640757272712871715001634380\ 1536321138792611997536476738529202786951942992689202497300809859370152505593208\ 9257126000874667664191637426101190291630419146176593556580296055*2.618033988749\ 8948482045868343656381177203091798057628621354486227052604628189024497072072041\ 8939113748475408807538689175212663386222353693179318006076672635443338908659593\ 9582905638322661319928290267880675208766892501711696207032221043216269548626296\ 31361443814975870122034080588795445474924618569536^n*(1.50000000000000000000000\ 0000000000000000000000000000000000000000000000000000000000000000000000000000000\ 0000000000000000000000000000000000000000000000000000000000000000000000000000000\ 0000000000000000000000000000000000000000000000000000000000000000000000000000000\ 000000000000000000000000000000000000000*n+1.+.500000000000000000000000000000000\ 0000000000000000000000000000000000000000000000000000000000000000000000000000000\ 0000000000000000000000000000000000000000000000000000000000000000000000000000000\ 0000000000000000000000000000000000000000000000000000000000000000000000000000000\ 000000000000000000000000000000*n^2) The BZ constant, in decimals, is .260990336999411149945686272447573459233731406912272027958512287149458591785853\ 7183603603434605903699146228932221665938119091496284518068180418083402936675848\ 2697888493726166427157460690339159842550013178116909459800958549875938043784201\ 1109472307259356240853638648726833908512972544946595778639894042 Theorem number, 2 Part I: Let a(n) be the number of spanning trees of the graph, G[n, 3], whose vertice\ s are 1, ...,n, arranged in a circle and there is an edge between two v\ ertices iff their distane is <=, 3, Then infinity ----- \ n 17 16 15 14 ) a[n + 3] x = - (3072 x - 11683 x - 26868 x - 60636 x / ----- n = 0 13 12 11 10 9 + 356682 x + 844329 x + 1651344 x + 104646 x - 813834 x 8 7 6 5 4 - 3128248 x - 1452330 x - 512250 x + 1392528 x + 1049445 x 3 2 / 2 + 579514 x + 54068 x - 15716 x - 16807) / ((x - 1) / 4 3 2 2 4 3 2 2 (x + 3 x + 6 x + 3 x + 1) (x - 4 x - x - 4 x + 1) ) and in Maple notation -(3072*x^17-11683*x^16-26868*x^15-60636*x^14+356682*x^13+844329*x^12+1651344*x^ 11+104646*x^10-813834*x^9-3128248*x^8-1452330*x^7-512250*x^6+1392528*x^5+ 1049445*x^4+579514*x^3+54068*x^2-15716*x-16807)/(x-1)^2/(x^4+3*x^3+6*x^2+3*x+1) ^2/(x^4-4*x^3-x^2-4*x+1)^2 The asympotics, in decimals, is 2352.17931274199798580466638294441235414677776296619526114354496566462012021505\ 2851608864350886562709976200566151925310547380485815738705353659214615395309821\ 2610710790374241382860497078697217740304380818043015830463297729894994193790514\ 8081723424643786203557608606406512266288587965817562946727784474*4.419480365787\ 5667074848413155156881333186325196333651926684832652318194029734233993516067056\ 2235617294200632046839265454646838409371545219466019855889276199478293020635623\ 0468040676060014431004669121124061629898241290300356640072093270969917949814978\ 61668038627041193408024058104919310419706116006054^n*(1.+n) Part II: Let b(n) be the sum of the number of leaves in all spanning trees of the abo\ ve-mentioned graph, G[n, 3], Then infinity ----- \ n 26 25 24 23 ) b[n + 3] x = (-8820 x + 51390 x + 61812 x + 2088 x / ----- n = 0 22 21 20 19 18 - 2539950 x - 2981160 x + 2492784 x + 45845688 x + 83018808 x 17 16 15 14 + 107694630 x - 44892840 x - 166389300 x - 333210654 x 13 12 11 10 - 121438506 x + 42702660 x + 312824052 x + 213402930 x 9 8 7 6 5 + 100784592 x - 77616756 x - 90041700 x - 62209728 x - 13836186 x 4 3 2 / 3 + 276924 x + 2761596 x + 501534 x + 32592 x - 54432) / ((x - 1) / 4 3 2 3 4 3 2 3 (x + 3 x + 6 x + 3 x + 1) (x - 4 x - x - 4 x + 1) ) and in Maple notation (-8820*x^26+51390*x^25+61812*x^24+2088*x^23-2539950*x^22-2981160*x^21+2492784*x ^20+45845688*x^19+83018808*x^18+107694630*x^17-44892840*x^16-166389300*x^15-\ 333210654*x^14-121438506*x^13+42702660*x^12+312824052*x^11+213402930*x^10+ 100784592*x^9-77616756*x^8-90041700*x^7-62209728*x^6-13836186*x^5+276924*x^4+ 2761596*x^3+501534*x^2+32592*x-54432)/(x-1)^3/(x^4+3*x^3+6*x^2+3*x+1)^3/(x^4-4* x^3-x^2-4*x+1)^3 The asympotics, in decimals, is 1462.63023213410584697035705538455286776446411245382557294257106633324402673727\ 4461260469652787358529673983689736946832460866512110438083053749494848431458425\ 8948306001765863828555822918721449022676839985547334087414132339080140917536001\ 8824443318186495330681368455243962328496428386929817587145091488*4.419480365787\ 5667074848413155156881333186325196333651926684832652318194029734233993516067056\ 2235617294200632046839265454646838409371545219466019855889276199478293020635623\ 0468040676060014431004669121124061629898241290300356640072093270969917949814978\ 61668038627041193408024058104919310419706116006054^n*(1.50000000000000000000000\ 0000000000000000000000000000000000000000000000000000000000000000000000000000000\ 0000000000000000000000000000000000000000000000000000000000000000000000000000000\ 0000000000000000000000000000000000000000000000000000000000000000000000000000000\ 000000000000000000000000000000000000000*n+1.+.500000000000000000000000000000000\ 0000000000000000000000000000000000000000000000000000000000000000000000000000000\ 0000000000000000000000000000000000000000000000000000000000000000000000000000000\ 0000000000000000000000000000000000000000000000000000000000000000000000000000000\ 000000000000000000000000000000*n^2) The BZ constant, in decimals, is .310909594394204344008503510728871359457534156622455497862965574296392568706818\ 2389717542663389754515614209307854901398957824822098178113415350516967063041587\ 8171787527146830199727656701613986631861076178492467710995347917371238944954205\ 2891443925255860750442024536049386427251672816518078123146621503 Theorem number, 3 Part I: Let a(n) be the number of spanning trees of the graph, G[n, 4], whose vertice\ s are 1, ...,n, arranged in a circle and there is an edge between two v\ ertices iff their distane is <=, 4, Then infinity ----- \ n 53 52 51 50 ) a[n + 4] x = - (640000 x - 3750617 x - 10123230 x - 1417153 x / ----- n = 0 49 48 47 46 - 19407892 x + 1740647769 x + 3664525174 x + 1281378322 x 45 44 43 42 + 1474725222 x - 130425774277 x - 161571579652 x + 276643639983 x 41 40 39 - 127937050526 x + 2049991617040 x + 4147938524334 x 38 37 36 - 6914988457309 x - 2171610912496 x - 9644939666019 x 35 34 33 - 36557666838062 x + 52531258489387 x + 37111231396788 x 32 31 30 + 16212599809187 x + 108679784757746 x - 223245543448335 x 29 28 27 - 50901893595560 x - 11312448949745 x - 84184506987858 x 26 25 24 + 367100960860495 x - 31301880304088 x - 8059995095637 x 23 22 21 - 20895465258994 x - 241462977425383 x + 72647169201204 x 20 19 18 + 20906822167189 x + 32766227152030 x + 62314593399081 x 17 16 15 - 26501665160740 x - 12977146508401 x - 3045687897606 x 14 13 12 - 7952556213263 x + 2756400915976 x + 2447122755810 x 11 10 9 8 + 177010550072 x + 328728888428 x - 98055106776 x - 144540407564 x 7 6 5 4 - 18882720344 x - 2102598597 x + 3137317214 x + 2213563334 x 3 2 / 2 + 290464530 x + 42522964 x - 4070186 x - 4782969) / ((x + 1) / 6 5 4 3 2 2 (x - 3 x + 6 x - 10 x + 6 x - 3 x + 1) 8 7 6 5 4 3 2 2 12 11 (x - 4 x - 17 x + 8 x + 49 x + 8 x - 17 x - 4 x + 1) (x + 3 x 10 9 8 7 6 5 4 3 2 + 12 x + 28 x - 27 x + 36 x - 81 x + 36 x - 27 x + 28 x + 12 x 2 + 3 x + 1) ) and in Maple notation -(640000*x^53-3750617*x^52-10123230*x^51-1417153*x^50-19407892*x^49+1740647769* x^48+3664525174*x^47+1281378322*x^46+1474725222*x^45-130425774277*x^44-\ 161571579652*x^43+276643639983*x^42-127937050526*x^41+2049991617040*x^40+ 4147938524334*x^39-6914988457309*x^38-2171610912496*x^37-9644939666019*x^36-\ 36557666838062*x^35+52531258489387*x^34+37111231396788*x^33+16212599809187*x^32 +108679784757746*x^31-223245543448335*x^30-50901893595560*x^29-11312448949745*x ^28-84184506987858*x^27+367100960860495*x^26-31301880304088*x^25-8059995095637* x^24-20895465258994*x^23-241462977425383*x^22+72647169201204*x^21+ 20906822167189*x^20+32766227152030*x^19+62314593399081*x^18-26501665160740*x^17 -12977146508401*x^16-3045687897606*x^15-7952556213263*x^14+2756400915976*x^13+ 2447122755810*x^12+177010550072*x^11+328728888428*x^10-98055106776*x^9-\ 144540407564*x^8-18882720344*x^7-2102598597*x^6+3137317214*x^5+2213563334*x^4+ 290464530*x^3+42522964*x^2-4070186*x-4782969)/(x+1)^2/(x^6-3*x^5+6*x^4-10*x^3+6 *x^2-3*x+1)^2/(x^8-4*x^7-17*x^6+8*x^5+49*x^4+8*x^3-17*x^2-4*x+1)^2/(x^12+3*x^11 +12*x^10+28*x^9-27*x^8+36*x^7-81*x^6+36*x^5-27*x^4+28*x^3+12*x^2+3*x+1)^2 The asympotics, in decimals, is 519711.426852910537863373535313590416475735909587413307729995199002085178649977\ 1051361813183394971342066886102756725669072217167905760433758963445683138222974\ 6959027978654677328529385303447547890156662034070431876816045816223817956166754\ 5750964700863444341252813623554889866564690956786580524519389877*6.298096056225\ 7735736763280347455186344983566539488734555762957580607021719381702047684955966\ 0868174082297905849451741105870880197908448107777793195646538809910654441251005\ 5905938348446142400780573792080060295441158235077834737768247256185377889131512\ 94989027077865744774017135367553226025262757348097^n*(1.+n) Part II: Let b(n) be the sum of the number of leaves in all spanning trees of the abo\ ve-mentioned graph, G[n, 4], Then infinity ----- \ n 80 79 78 ) b[n + 4] x = (-2286144 x + 20288352 x + 23120304 x / ----- n = 0 77 76 75 74 - 182262896 x - 413789616 x - 11183292912 x + 632997952 x 73 72 71 + 128235780288 x + 394063035264 x + 2278879967520 x 70 69 68 - 78892093920 x - 16437787090944 x - 29386270966384 x 67 66 65 - 128520156090960 x + 61725157571712 x + 945567665234912 x 64 63 62 + 392081004388080 x + 2358150010092432 x + 410192893927872 x 61 60 59 - 25198973087325216 x - 5457673124694048 x + 943748361980992 x 58 57 56 - 33571674940095072 x + 304407133421790048 x + 124307542565813968 x 55 54 - 354351070785577968 x + 281590846668744912 x 53 52 - 1869525145677951504 x - 978632778690640416 x 51 50 + 3303116736536515008 x - 955469602589286832 x 49 48 + 6872242222937137104 x + 1812807126628204944 x 47 46 - 14522552017154658832 x + 4447125213438707280 x 45 44 - 15930483996144874608 x + 5848895353422382416 x 43 42 + 29370448454775059568 x - 12861273847005167952 x 41 40 + 20895849697931029744 x - 22473411015688405344 x 39 38 - 27906452777063436096 x + 18974217157259796688 x 37 36 - 14193574889032233360 x + 26876299571644542480 x 35 34 + 11222080290298611216 x - 15057923609939951424 x 33 32 + 4391704317367694784 x - 14218634762755444384 x 31 30 - 836993539351119744 x + 6880225137184026912 x 29 28 - 508872690132670912 x + 3372060370229907408 x 27 26 25 - 320102087347832976 x - 1944754948018462464 x + 82097742232158240 x 24 23 22 - 362288931342079440 x + 40606459891624144 x + 314132185259406720 x 21 20 19 + 5712495177821280 x + 2704411631741632 x - 1931352341682528 x 18 17 16 - 25604092681777440 x - 3104859175196256 x + 1900120616570832 x 15 14 13 + 508006500456144 x + 1012333704242480 x + 206779130325648 x 12 11 10 - 98956501771200 x - 39815987534944 x - 21825432700560 x 9 8 7 6 - 3175928051472 x + 1852842506256 x + 617264395248 x + 211863823152 x 5 4 3 2 + 29436460912 x - 7790063472 x - 1409690064 x - 390663920 x / 3 - 29842992 x + 18874368) / ((x + 1) / 6 5 4 3 2 3 (x - 3 x + 6 x - 10 x + 6 x - 3 x + 1) 8 7 6 5 4 3 2 3 12 11 (x - 4 x - 17 x + 8 x + 49 x + 8 x - 17 x - 4 x + 1) (x + 3 x 10 9 8 7 6 5 4 3 2 + 12 x + 28 x - 27 x + 36 x - 81 x + 36 x - 27 x + 28 x + 12 x 3 + 3 x + 1) ) and in Maple notation (-2286144*x^80+20288352*x^79+23120304*x^78-182262896*x^77-413789616*x^76-\ 11183292912*x^75+632997952*x^74+128235780288*x^73+394063035264*x^72+ 2278879967520*x^71-78892093920*x^70-16437787090944*x^69-29386270966384*x^68-\ 128520156090960*x^67+61725157571712*x^66+945567665234912*x^65+392081004388080*x ^64+2358150010092432*x^63+410192893927872*x^62-25198973087325216*x^61-\ 5457673124694048*x^60+943748361980992*x^59-33571674940095072*x^58+ 304407133421790048*x^57+124307542565813968*x^56-354351070785577968*x^55+ 281590846668744912*x^54-1869525145677951504*x^53-978632778690640416*x^52+ 3303116736536515008*x^51-955469602589286832*x^50+6872242222937137104*x^49+ 1812807126628204944*x^48-14522552017154658832*x^47+4447125213438707280*x^46-\ 15930483996144874608*x^45+5848895353422382416*x^44+29370448454775059568*x^43-\ 12861273847005167952*x^42+20895849697931029744*x^41-22473411015688405344*x^40-\ 27906452777063436096*x^39+18974217157259796688*x^38-14193574889032233360*x^37+ 26876299571644542480*x^36+11222080290298611216*x^35-15057923609939951424*x^34+ 4391704317367694784*x^33-14218634762755444384*x^32-836993539351119744*x^31+ 6880225137184026912*x^30-508872690132670912*x^29+3372060370229907408*x^28-\ 320102087347832976*x^27-1944754948018462464*x^26+82097742232158240*x^25-\ 362288931342079440*x^24+40606459891624144*x^23+314132185259406720*x^22+ 5712495177821280*x^21+2704411631741632*x^20-1931352341682528*x^19-\ 25604092681777440*x^18-3104859175196256*x^17+1900120616570832*x^16+ 508006500456144*x^15+1012333704242480*x^14+206779130325648*x^13-98956501771200* x^12-39815987534944*x^11-21825432700560*x^10-3175928051472*x^9+1852842506256*x^ 8+617264395248*x^7+211863823152*x^6+29436460912*x^5-7790063472*x^4-1409690064*x ^3-390663920*x^2-29842992*x+18874368)/(x+1)^3/(x^6-3*x^5+6*x^4-10*x^3+6*x^2-3*x +1)^3/(x^8-4*x^7-17*x^6+8*x^5+49*x^4+8*x^3-17*x^2-4*x+1)^3/(x^12+3*x^11+12*x^10 +28*x^9-27*x^8+36*x^7-81*x^6+36*x^5-27*x^4+28*x^3+12*x^2+3*x+1)^3 The asympotics, in decimals, is 343276.824817876495532581035532414579881522059402660666801636912795735480721939\ 3387469818623378037822333916801769221032067352684690750266193923762693604100404\ 9463689724280403505554258169509815064035702623548060417291463730064412744434328\ 7202429581101939105708235379035273551732697632885572073029789531*6.298096056225\ 7735736763280347455186344983566539488734555762957580607021719381702047684955966\ 0868174082297905849451741105870880197908448107777793195646538809910654441251005\ 5905938348446142400780573792080060295441158235077834737768247256185377889131512\ 94989027077865744774017135367553226025262757348097^n*(1.50000000000000000000000\ 0000000000000000000000000000000000000000000000000000000000000000000000000000000\ 0000000000000000000000000000000000000000000000000000000000000000000000000000000\ 0000000000000000000000000000000000000000000000000000000000000000000000000000000\ 000000000000000000000000000000000000000*n+1.+.500000000000000000000000000000000\ 0000000000000000000000000000000000000000000000000000000000000000000000000000000\ 0000000000000000000000000000000000000000000000000000000000000000000000000000000\ 0000000000000000000000000000000000000000000000000000000000000000000000000000000\ 000000000000000000000000000000*n^2) The BZ constant, in decimals, is .330257145678491294152201301215982579594511047966878645657116548401719850150894\ 8011993845958733621898749861243663556491188833571288442784842579638132249880330\ 3136653393885225076822939108298232248635475703397612277447225929476675876480347\ 1615233791622060126409183346294148549392148036565874334392134278 Theorem number, 4 Part I: Let a(n) be the number of spanning trees of the graph, G[n, 5], whose vertice\ s are 1, ...,n, arranged in a circle and there is an edge between two v\ ertices iff their distane is <=, 5, Then infinity ----- \ n 161 160 159 ) a[n + 5] x = - (248832000 x - 1963657584 x - 4194274728 x / ----- n = 0 158 157 156 + 2032496647 x - 75426080792 x - 284884355888 x 155 154 153 + 17020273884786 x + 32259840026256 x + 52460927780982 x 152 151 150 + 696146163811972 x + 725550654375496 x - 26219760321723941 x 149 148 147 - 9283813674332256 x + 49320071066548224 x - 794480244967747584 x 146 145 - 978964613497109549 x + 14521052864503430056 x 144 143 + 894177689230001812 x - 48252416822386418634 x 142 141 + 273043108453098689232 x + 572467080016761685482 x 140 139 - 3993851581139799487435 x - 266062009522643110720 x 138 137 + 14700762357308700384890 x - 45890407655248774916472 x 136 135 - 127650512869807009046928 x + 598213294456279482688854 x 134 133 - 10465999224552222981901 x - 1879101191014314192612616 x 132 131 + 3900257353683237367579172 x + 14414402528103229252833942 x 130 129 - 55298060156739954979558656 x + 20375877576112606731723918 x 128 127 + 111900276543887915404203788 x - 229214207805069894597390376 x 126 125 - 671520209965461953647972414 x + 2688823143195025638484277832 x 124 123 - 1731541133831635370426702448 x - 3330493668032112602099501094 x 122 121 + 8246933357326057817312367303 x + 16679812588976770321032903024 x 120 119 - 73918856527950114038384975157 x + 59462617356214134570121184670 x 118 117 + 50455463720316492163276123536 x - 174013931216313370420658908422 x 116 - 251653695492186053842626957480 x 115 + 1257099136018116260910239390664 x 114 - 1068308614221141563927385498594 x 113 - 371784843893001450112575887238 x 112 + 2135147974459460383649585688384 x 111 + 2583158624525409687892437631626 x 110 - 13851329107415923977263314050696 x 109 + 10999205756329419964361816342520 x 108 + 550788013954062846351670861746 x 107 - 15439961455946054213464964126298 x 106 - 19933210509524749011298907239056 x 105 + 102864504755213071945644354921906 x 104 - 62409302806638304975781569039640 x 103 + 13363752087042160124337333512008 x 102 + 57718111815162451256109766308598 x 101 + 102691382446613825578916562346338 x 100 - 528755976520196627631061929512832 x 99 + 178524412821349876422364334021658 x 98 - 98007390649768215617561091382202 x 97 - 23568330209040119006174708003384 x 96 - 297869164308437820228923148333494 x 95 + 1794747224561943114760694511813402 x 94 - 229574708671509279525062897116176 x 93 + 410017522318052489351696508109974 x 92 - 379534144914488212623685088132480 x 91 + 370010046364659615633890243930968 x 90 - 4184803547786165819150271564446012 x 89 - 30808011499154554851247603237014 x 88 - 1055679249696494520772867746483072 x 87 + 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/(x^32+x^31+12*x^30+45*x^29+45*x^28-1561*x^27+3917*x^26-3222*x^25-3981*x^24+ 7745*x^23+26379*x^22-88937*x^21+84093*x^20+63864*x^19-153881*x^18-202281*x^17+ 550163*x^16-202281*x^15-153881*x^14+63864*x^13+84093*x^12-88937*x^11+26379*x^10 +7745*x^9-3981*x^8-3222*x^7+3917*x^6-1561*x^5+45*x^4+45*x^3+12*x^2+x+1)^2 The asympotics, in decimals, is 209440027.172205912557304400855939676227695051321648580552970335788953228699383\ 4144011647994763413220712077145768006761245691339339957251985341175824557159397\ 4947274772308799597391558419656532460301294999659114343157865433052551355135614\ 7212102806510906826384473664102993695295197889367171565990546393*8.216271114701\ 8080784197270517005864370282025075930083403333260117064480028303210164396856662\ 1743991959463897966104229258622840833148397780121598105156859723905359918565442\ 5456249668868065723070529646912028862036409706909970198249278968640386785790148\ 21722529481650260900175511656134298623976970864339^n*(1.+n) Part II: Let b(n) be the sum of the number of leaves in all spanning trees of the abo\ ve-mentioned graph, G[n, 5], Then infinity ----- \ n 242 241 240 ) b[n + 5] x = (-1067328900 x + 12702084750 x + 1259570200 x / ----- n = 0 239 238 237 - 152499860250 x + 125762465850 x - 1537327071400 x 236 235 234 - 132527916507270 x + 126333486185040 x + 1615090490177600 x 233 232 231 + 2288407326051990 x + 56976560450775240 x + 552645211806817360 x 230 229 - 298167590246735690 x - 2080773463021646670 x 228 227 + 8073454926148285200 x - 100335877991615756480 x 226 225 - 748116738540137738430 x + 1058336370691848429920 x 224 223 + 1587818685785421975360 x - 22007914431900086782470 x 222 221 + 76770596902166848986360 x + 552782694816515200028310 x 220 219 - 1005250653533165220003570 x - 1320933779281277374870280 x 218 217 + 16339555806755936769309150 x - 26220545678930735482834650 x 216 215 - 251164542673236234739538920 x + 435315480678773723083675390 x 214 213 + 762858271257235141645910310 x - 6109238851442820785445361320 x 212 211 + 4088816776786875017898293430 x + 71698413436419824745383096580 x 210 - 108534858079360427148406305520 x 209 - 235347730051274802356402192350 x 208 + 1328827083247699906008238118340 x 207 - 124102153297988970903984695600 x 206 - 13253442713374281203628016005250 x 205 + 17799320923245542414136099466230 x 204 + 41066263404419950413725726905760 x 203 - 184272224443111432369154536189460 x 202 - 39959677438010569110570068550750 x 201 + 1619087030800343003128378804760640 x 200 - 2119456381719368964412190696734000 x 199 - 4077668792857025845453473778671150 x 198 + 16826736590205026242446207764157880 x 197 + 5432278741607025173877081421629270 x 196 - 134023057667117314161016853419053510 x 195 + 190345470753034569376277900700075720 x 194 + 221472024327534993831436105105207930 x 193 - 1047155731082677225576318804021617570 x 192 - 169026294513701583687225528717592520 x 191 + 7217861246702823604002894208163263290 x 190 - 11529972221495942056549603231186908390 x 189 - 6095165643785192302444014610424893400 x 188 + 43987284069000273043451993973265634010 x 187 - 4579251547363662992016243758619849840 x 186 - 257458122500838143376056861007918907280 x 185 + 459083676198047111630413506151451986150 x 184 + 43598899488260135561582528689849094820 x 183 - 1252334423812000184763403758111503497040 x 182 + 460052040126623836239953767810952766430 x 181 + 6342344120397932122223983619405638461990 x 180 - 12358781288314789553509220147081863918000 x 179 + 2387087992687599243760590647640552818820 x 178 + 24558895747340836078622168516438766143010 x 177 - 14526249445455966617946125621789508511200 x 176 - 112484769989980537999047133707588669737940 x 175 + 232691717874048065685995443052726080713810 x 174 - 88890080490045230915313801035456850863640 x 173 - 337571578537659099667807791524875467827850 x 172 + 257945752175720417678777860624345020280350 x 171 + 1493201539523349084002079759331601771371080 x 170 - 3144599091806574828509056262176493924653850 x 169 + 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3133798474023779543635053803933295829951040741911140115928274323589448047421785\ 4074469195763126131495984210771892287197387049620101040090196611*8.216271114701\ 8080784197270517005864370282025075930083403333260117064480028303210164396856662\ 1743991959463897966104229258622840833148397780121598105156859723905359918565442\ 5456249668868065723070529646912028862036409706909970198249278968640386785790148\ 21722529481650260900175511656134298623976970864339^n*(1.50000000000000000000000\ 0000000000000000000000000000000000000000000000000000000000000000000000000000000\ 0000000000000000000000000000000000000000000000000000000000000000000000000000000\ 0000000000000000000000000000000000000000000000000000000000000000000000000000000\ 000000000000000000000000000000000000000*n+1.+.500000000000000000000000000000000\ 0000000000000000000000000000000000000000000000000000000000000000000000000000000\ 0000000000000000000000000000000000000000000000000000000000000000000000000000000\ 0000000000000000000000000000000000000000000000000000000000000000000000000000000\ 000000000000000000000000000000*n^2) The BZ constant, in decimals, is .340179868924112836772060183916870487107478540491420808722455473554087114986963\ 7106305885610306541613274353498978178840660433379254041238701241735752591276975\ 0644465452717178780407367931860440780427263450567398621830464946602601346845038\ 5989127893935952397101240552277896176385243683381867509462708192 -------------------------------- This took, 3538.936, seconds. 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