Generating Functions for Enumerating the Number of Spanning Trees in Friends\ hip graphs where n people live in a one-sided CIRCULAR street, and every\ one is friends with all the neighbors distance at most r for r from 2 to, 5 By Shalosh B. Ekhad Theorem number, 1 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 distance is <=, 2, Then infinity ----- 5 4 3 2 \ n 36 x - 132 x - 46 x + 353 x + 116 x - 125 ) a[n + 5] 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 Theorem number, 2 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 distance is <=, 3, Then infinity ----- \ n 17 16 15 14 ) a[n + 7] 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 Theorem number, 3 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 distance is <=, 4, Then infinity ----- \ n 53 52 51 50 ) a[n + 9] 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 Theorem number, 4 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 distance is <=, 5, Then infinity ----- \ n 161 160 159 ) a[n + 11] 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 + 1107024719169772753422009607571946 x 86 - 75000007032475436428076064064978 x 85 + 6999078253052927478788111778058184 x 84 + 566717851984974334583451481048586 x 83 + 1748022887671391426612499603146130 x 82 - 1607515438127554142752107250326960 x 81 - 150925418900405736523131085643874 x 80 - 8346137850651287231466149263910974 x 79 - 882911530071755825743607168107576 x 78 - 1896675318016809937790004284181334 x 77 + 1477408490483544262753957873043106 x 76 - 33199575128343087875798703757056 x 75 + 7003765793932462368309531315177150 x 74 + 565266616576537189951271871475646 x 73 + 1285773701741464762931863012719080 x 72 - 910767970071720206098789882015006 x 71 + 324850785645941464890162113501658 x 70 - 4189928736714587984356704609960816 x 69 - 43482095528646801338198296632090 x 68 - 430329762102200076954596658730424 x 67 + 387394888810491489841871286480184 x 66 - 382351663024493509568612053616558 x 65 + 1785767160749773297608065783442642 x 64 - 101659696421413382493359247116064 x 63 - 23336467112707963462897704784674 x 62 - 119688557143595674140667727106602 x 61 + 222868913428352755878056583749656 x 60 - 515806422701247975330723137537408 x 59 + 40544357833186052498076267565458 x 58 + 60497691069399340993212253893264 x 57 + 26659041003944107647213165923742 x 56 - 70650439602997268636407891602992 x 55 + 97386448129189825492019941697560 x 54 - 7718852297739146772089990761598 x 53 - 16306841556085076958453543339882 x 52 - 2406595660843151850193165491744 x 51 + 12234797105942235569405518147422 x 50 - 12807737662096956486922758384960 x 49 48 + 966737707693205690187195105768 x + 2276476353422989193961258964014 x 47 46 - 9389264409194265704408706582 x - 1204935328651988679803582192976 x 45 44 + 1153987811415912950574780880758 x - 108062476548546606546519937152 x 43 42 - 190863615529928672225110296168 x + 23321755680001034664735531402 x 41 40 + 69950405834356985856983020110 x - 68156177778708080091371218800 x 39 38 + 8420712707315106012134480598 x + 9501367157391272958927104715 x 37 36 - 2112249158282059145986036944 x - 2249427931822604975550638490 x 35 34 + 2518200413299028555614082676 x - 374543384176086965857979232 x 33 32 - 281235627884111453627006676 x + 79174398517080720020435108 x 31 30 + 34876830335155008477364544 x - 53094389244461298349829503 x 29 28 + 8253638847757506310132998 x + 4983957519313664051410848 x 27 26 - 1337432449160541218448642 x - 218822934774110700936745 x 25 24 + 588730461352364865014384 x - 59072210059642063080190 x 23 22 - 53733631959181055774616 x + 8922529109102454087168 x 21 20 + 1116646465912310695440 x - 3941108835166048697647 x 19 18 + 117196645913298608984 x + 290145971426701996604 x 17 16 - 17158139216674293114 x - 2453189726804992848 x 15 14 + 14415563168531272260 x + 664300906124185063 x 13 12 11 - 725871940133709824 x - 26943639129911366 x - 9522586356425952 x 10 9 8 - 27549131432584944 x - 2332231658845500 x + 436372250830996 x 7 6 5 + 93364112005072 x + 40563010269100 x + 21817120471590 x 4 3 2 + 2095363792560 x + 152840200848 x + 20396723899 x - 1872418472 x / 2 - 2357947691) / ((x - 1) / 8 7 6 5 4 3 2 2 (x + 3 x + 6 x + 10 x + 15 x + 10 x + 6 x + 3 x + 1) 8 7 6 5 4 3 2 2 16 15 (x + 3 x + 6 x - x + 15 x - x + 6 x + 3 x + 1) (x - 5 x 14 13 12 11 10 9 8 7 + 10 x - 10 x - 28 x + 10 x + 110 x + 110 x + 88 x + 110 x 6 5 4 3 2 2 16 15 14 + 110 x + 10 x - 28 x - 10 x + 10 x - 5 x + 1) (x - 5 x - 23 x 13 12 11 10 9 8 7 6 - 10 x - 94 x - 485 x + 242 x + 110 x + 649 x + 110 x + 242 x 5 4 3 2 2 32 31 30 29 - 485 x - 94 x - 10 x - 23 x - 5 x + 1) (x + x + 12 x + 45 x 28 27 26 25 24 23 + 45 x - 1561 x + 3917 x - 3222 x - 3981 x + 7745 x 22 21 20 19 18 17 + 26379 x - 88937 x + 84093 x + 63864 x - 153881 x - 202281 x 16 15 14 13 12 11 + 550163 x - 202281 x - 153881 x + 63864 x + 84093 x - 88937 x 10 9 8 7 6 5 4 + 26379 x + 7745 x - 3981 x - 3222 x + 3917 x - 1561 x + 45 x 3 2 2 + 45 x + 12 x + x + 1) ) and in Maple notation -(248832000*x^161-1963657584*x^160-4194274728*x^159+2032496647*x^158-\ 75426080792*x^157-284884355888*x^156+17020273884786*x^155+32259840026256*x^154+ 52460927780982*x^153+696146163811972*x^152+725550654375496*x^151-\ 26219760321723941*x^150-9283813674332256*x^149+49320071066548224*x^148-\ 794480244967747584*x^147-978964613497109549*x^146+14521052864503430056*x^145+ 894177689230001812*x^144-48252416822386418634*x^143+273043108453098689232*x^142 +572467080016761685482*x^141-3993851581139799487435*x^140-266062009522643110720 *x^139+14700762357308700384890*x^138-45890407655248774916472*x^137-\ 127650512869807009046928*x^136+598213294456279482688854*x^135-\ 10465999224552222981901*x^134-1879101191014314192612616*x^133+ 3900257353683237367579172*x^132+14414402528103229252833942*x^131-\ 55298060156739954979558656*x^130+20375877576112606731723918*x^129+ 111900276543887915404203788*x^128-229214207805069894597390376*x^127-\ 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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 -------------------------------- This took, 538.169, seconds. 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