> # This Maple (version 5.0) work sheet is a supplement to > # Czedli, Ozsvart and Udvari's paper: > # " Counting lattices. I. Slim semimodular lattices " > # March 29, 2011 > restart; > nnmax:=100; # Rewrite if you like but be careful since you may exceed the capacity of your computer. > dstep:=-2010: # Do not rewrite > nnmax := 100 > > arrbifactorial:=array(-1..nnmax): > bifactorial:=proc(j) local i,produc: produc:=1; for i from 1 by 2 to 2*j-1 do > produc:=produc*i od; produc end: # yields 1*3*5*7* ... *(2*j-1). > for j from -1 to nnmax do arrbifactorial[j]:=bifactorial(j) od: > # > #The next procedures follow the notation of the paper. > # > arrs0:=array(1..nnmax,1..nnmax,0..nnmax): > for m from 1 to nnmax do for n from m to nnmax do for k from 0 to m-1 do arrs0[m,n,k]:=dstep od od od: > s0:=proc(m,n,k) global arrs0: if not (0<= k and k if arrs0[m,n,k]>dstep then arrs0[m,n,k] > else > if m=1 or k=0 then 1 > else arrs0[m,n,k]:=binomial(m-1,k)*n!/((n-k)!)-binomial(m-2,k-1)*n!/((n-k+1)!) > fi > fi end: > # > s:=proc(m,n,k): s0(m,n,k)+s1(m,n,k) end: > # > > arrs1:=array(1..nnmax,1..nnmax,0..nnmax): > for m from 1 to nnmax do for n from m to nnmax do for k from 0 to m-1 do arrs1[m,n,k]:=dstep od od od: > s1:=proc(m,n,k) global arrs1: local summa,j: if not (0<= k and k if arrs1[m,n,k]>dstep then arrs1[m,n,k] > else summa:=0; > if k=0 then 0 > else for j from 0 by 1 to k-1 do summa:=summa+j!*s(m-j-1,n-j-1,k-j-1)*(n-j-2) od; arrs1[m,n,k]:=summa > fi; > fi; end: > # > arrz0:=array(1..nnmax, 0..nnmax): for m from 1 to nnmax do for k from 0 to m-1 do arrz0[m,k]:=dstep od od: > z0:=proc(m,k) global arrz0; local summa, j: if not (0<= k and k if arrz0[m,k]>dstep then arrz0[m,k] > else summa:=0; for j from 0 to floor(k/2) do summa:=summa+ bifactorial(j)*binomial(k,k-2*j) od; > summa:=binomial(m-1,k)*summa; arrz0[m,k]:=summa > fi end: > > # > z:=proc(m,k): z0(m,k)+z1(m,k) end: > # > arrsymper:=array(-1..nnmax): for i from -1 to nnmax do arrsymper[i]:=dstep od: > symper:=proc(i) global arrsymper; local summa,r: > if arrsymper[i]> dstep then arrsymper[i] > else summa:=0; for r from 0 by 1 to floor(i/2) do summa:=summa+binomial(i,i-2*r)*bifactorial(r) od; > arrsymper[i]:=summa; > fi; end: # This yields the last factor of the formula for z1 > # > # > arrz1:=array(1..nnmax, 0..nnmax): for m from 1 to nnmax do for k from 0 to m-1 do arrz1[m,k]:=dstep od od: > z1:=proc(m,k) global arrz1: local summa,i: if not (0<= k and k if arrz1[m,k]>dstep then arrz1[m,k] > else summa:=0; if k<2 then summa:=0 else > for i from 0 by 1 to k-2 do summa:=summa+(m-3-i)*z(m-2-i,k-2-i)*symper(i) od; > fi; arrz1[m,k]:=summa > fi; end: > # > > sstar:=proc(m,n,k): if m=n then (s(m,n,k)+z(m,k))/2 else s(m,n,k) fi end: > # > arrf:=array(0..nnmax): for i from 0 to nnmax do arrf[i]:=dstep od: > f:=proc(h) global arrf: local summa,k,n: if not (0<= h and h<=nnmax ) then print(ERROR7) fi; > if arrf[h]>dstep then arrf[h] > else summa:=0; for k from 0 by 1 to h-2 do > for n from ceil((h+k)/2) by 1 to h-1 do summa:=summa+sstar(h+k-n,n,k) od: od: arrf[h]:=summa > fi; end: > # f(h) is the number of indecomposable slim semimodular lattices of length h > # > > arrg:=array(0..nnmax): for i from 0 to nnmax do arrg[i]:=dstep od: > g:=proc(h) global arrg: local j,summa: summa:=0; if not (0<= h and h<=nnmax ) then print(ERROR8) fi; > if arrg[h]>dstep then arrg[h] > else if h<2 then arrg[h]:=1 > else for j from 2 to h do summa:=summa+f(j)*g(h-j) od; arrg[h]:=summa+g(h-1) > fi; > fi; end: > # Now, the program is basically ready. You can enter commands like > # f(12); or > # g(12); or > # for h from 0 to 12 do print("h= ",h, " f(h)= ",f(h), " g(h)= ",g(h)); od; > # to display values or to produce a list of values. > # Since previously computed data are stored, if you are interested in the time needed by, say, > # computing g(50), then restart the program (from the beginning), do not count anything else, > # and do not display anything else (since displaying also consumes time). > # > # Below, see a list up to h=nnmax (which is 100): > # > for h from 0 to 100 do > print("h= ",h, " f(h)= ", f(h) , " g(h)= ", g(h) ); print(", which is exactly (for small h) or approximately (for large h): "); > print(" f(h)= ",evalf(f(h)), " g(h)= ", evalf(g(h))); print("******************"); > print("******************"); od; > "h= ", 0, " f(h)= ", 0, " g(h)= ", 1 ", which is exactly (for small h) or approximately (for large h)\ : " " f(h)= ", 0, " g(h)= ", 1. "******************" "******************" "h= ", 1, " f(h)= ", 0, " g(h)= ", 1 ", which is exactly (for small h) or approximately (for large h)\ : " " f(h)= ", 0, " g(h)= ", 1. "******************" "******************" "h= ", 2, " f(h)= ", 1, " g(h)= ", 2 ", which is exactly (for small h) or approximately (for large h)\ : " " f(h)= ", 1., " g(h)= ", 2. "******************" "******************" "h= ", 3, " f(h)= ", 2, " g(h)= ", 5 ", which is exactly (for small h) or approximately (for large h)\ : " " f(h)= ", 2., " g(h)= ", 5. "******************" "******************" "h= ", 4, " f(h)= ", 8, " g(h)= ", 17 ", which is exactly (for small h) or approximately (for large h)\ : " " f(h)= ", 8., " g(h)= ", 17. "******************" "******************" "h= ", 5, " f(h)= ", 39, " g(h)= ", 73 ", which is exactly (for small h) or approximately (for large h)\ : " " f(h)= ", 39., " g(h)= ", 73. "******************" "******************" "h= ", 6, " f(h)= ", 242, " g(h)= ", 397 ", which is exactly (for small h) or approximately (for large h)\ : " " f(h)= ", 242., " g(h)= ", 397. "******************" "******************" "h= ", 7, " f(h)= ", 1759, " g(h)= ", 2623 ", which is exactly (for small h) or approximately (for large h)\ : " " f(h)= ", 1759., " g(h)= ", 2623. "******************" "******************" "h= ", 8, " f(h)= ", 14674, " g(h)= ", 20414 ", which is exactly (for small h) or approximately (for large h)\ : " " f(h)= ", 14674., " g(h)= ", 20414. "******************" "******************" "h= ", 9, " f(h)= ", 137127, " g(h)= ", 181607 ", which is exactly (for small h) or approximately (for large h)\ : " " f(h)= ", 137127., " g(h)= ", 181607. "******************" "******************" "h= ", 10, " f(h)= ", 1416430, " g(h)= ", 1809104 ", which is exactly (for small h) or approximately (for large h)\ : " 7 7 " f(h)= ", .1416430 10 , " g(h)= ", .1809104 10 "******************" "******************" "h= ", 11, " f(h)= ", 16006403, " g(h)= ", 19886032 ", which is exactly (for small h) or approximately (for large h)\ : " 8 8 " f(h)= ", .16006403 10 , " g(h)= ", .19886032 10 "******************" "******************" "h= ", 12, " f(h)= ", 196400810, " g(h)= ", 238723606 ", which is exactly (for small h) or approximately (for large h)\ : " 9 9 " f(h)= ", .196400810 10 , " g(h)= ", .238723606 10 "******************" "******************" "h= ", 13, " f(h)= ", 2600652663, " g(h)= ", 3105360727 ", which is exactly (for small h) or approximately (for large h)\ : " 10 10 " f(h)= ", .2600652663 10 , " g(h)= ", .3105360727 10 "******************" "******************" "h= ", 14, " f(h)= ", 36972248222, " g(h)= ", 43500615626 ", which is exactly (for small h) or approximately (for large h)\ : " 11 11 " f(h)= ", .3697224822 10 , " g(h)= ", .4350061563 10 "******************" "******************" "h= ", 15, " f(h)= ", 561800545219, " g(h)= ", 652817464376 ", which is exactly (for small h) or approximately (for large h)\ : " 12 12 " f(h)= ", .5618005452 10 , " g(h)= ", .6528174644 10 "******************" "******************" "h= ", 16, " f(h)= ", 9088375297906, " g(h)= ", 10448823841373 ", which is exactly (for small h) or approximately (for large h)\ : " 13 14 " f(h)= ", .9088375298 10 , " g(h)= ", .1044882384 10 "******************" "******************" "h= ", 17, " f(h)= ", 155975625064575, " g(h)= ", 177677335375138 ", which is exactly (for small h) or approximately (for large h)\ : " 15 15 " f(h)= ", .1559756251 10 , " g(h)= ", .1776773354 10 "******************" "******************" "h= ", 18, " f(h)= ", 2830849642806358, " g(h)= ", 3198823376772421 ", which is exactly (for small h) or approximately (for large h)\ : " 16 16 " f(h)= ", .2830849643 10 , " g(h)= ", .3198823377 10 "******************" "******************" "h= ", 19, " f(h)= ", 54177933487381259, " g(h)= ", 60786586167252310 ", which is exactly (for small h) or approximately (for large h)\ : " 17 17 " f(h)= ", .5417793349 10 , " g(h)= ", .6078658617 10 "******************" "******************" "h= ", 20, " f(h)= ", 1090548467176337306, " g(h)= ", 1215866504166363857 ", which is exactly (for small h) or approximately (for large h)\ : " 19 19 " f(h)= ", .1090548467 10 , " g(h)= ", .1215866504 10 "******************" "******************" "h= ", 21, " f(h)= ", 23033326647184628247, " g(h)= ", 25535358624325974593 ", which is exactly (for small h) or approximately (for large h)\ : " 20 20 " f(h)= ", .2303332665 10 , " g(h)= ", .2553535862 10 "******************" "******************" "h= ", 22, " f(h)= ", 509352549399084896990, " g(h)= ", 561814753179297973112 ", which is exactly (for small h) or approximately (for large h)\ : " 21 21 " f(h)= ", .5093525494 10 , " g(h)= ", .5618147532 10 "******************" "******************" "h= ", 23, " f(h)= ", 11769815832190412413411, " g(h)= ", 12922406041937871896542 ", which is exactly (for small h) or approximately (for large h)\ : " 23 23 " f(h)= ", .1176981583 10 , " g(h)= ", .1292240604 10 "******************" "******************" "h= ", 24, " f(h)= ", 283673506583402571854530, " g(h)= ", 310150504415028661579522 ", which is exactly (for small h) or approximately (for large h)\ : " 24 24 " f(h)= ", .2836735066 10 , " g(h)= ", .3101505044 10 "******************" "******************" "h= ", 25, " f(h)= ", 7119278235635625192863439, " g(h)= ", 7754020303820035152018010 ", which is exactly (for small h) or approximately (for large h)\ : " 25 25 " f(h)= ", .7119278236 10 , " g(h)= ", .7754020304 10 "******************" "******************" "h= ", 26, " f(h)= ", 185757472854637391268571654, " g(h)= ", 201610005573756661040571491 ", which is exactly (for small h) or approximately (for large h)\ : " 27 27 " f(h)= ", .1857574729 10 , " g(h)= ", .2016100056 10 "******************" "******************" "h= ", 27, " f(h)= ", 5031806452923701175381733979, " g(h)= ", 5443592382115569513884321429 ", which is exactly (for small h) or approximately (for large h)\ : " 28 28 " f(h)= ", .5031806453 10 , " g(h)= ", .5443592382 10 "******************" "******************" "h= ", 28, " f(h)= ", 141314621327147635243392156650, " g(h)= ", 152423443023543723212128936301 ", which is exactly (for small h) or approximately (for large h)\ : " 30 30 " f(h)= ", .1413146213 10 , " g(h)= ", .1524234430 10 "******************" "******************" "h= ", 29, " f(h)= ", 4109544646633449697987660474695, " g(h)= ", 4420349589077957078249798173618 ", which is exactly (for small h) or approximately (for large h)\ : " 31 31 " f(h)= ", .4109544647 10 , " g(h)= ", .4420349589 10 "******************" "******************" "h= ", 30, " f(h)= ", 123605400489105985056979061101550, " g(h)= ", 132612263247210183811631647481788 ", which is exactly (for small h) or approximately (for large h)\ : " 33 33 " f(h)= ", .1236054005 10 , " g(h)= ", .1326122632 10 "******************" "******************" "h= ", 31, " f(h)= ", 3841001395249767949623643340177299, " g(h)= ", 4111027208413046971886241653870383 ", which is exactly (for small h) or approximately (for large h)\ : " 34 34 " f(h)= ", .3841001395 10 , " g(h)= ", .4111027208 10 "******************" "******************" "h= ", 32, " f(h)= ", 123188551413293279742205514541425554, " g(h)= ", 131554165909245769308575570969427959 ", which is exactly (for small h) or approximately (for large h)\ : " 36 36 " f(h)= ", .1231885514 10 , " g(h)= ", .1315541659 10 "******************" "******************" "h= ", 33, " f(h)= ", 4073779384608429955541427356094817215, " g(h)= ", 4341324466283448529484903614262655614 ", which is exactly (for small h) or approximately (for large h)\ : " 37 37 " f(h)= ", .4073779385 10 , " g(h)= ", .4341324466 10 "******************" "******************" "h= ", 34, " f(h)= ", 138781902598217032407444121690907429590, " g(h)= ", 147606126239848911222690178119813598031 ", which is exactly (for small h) or approximately (for large h)\ : " 39 39 " f(h)= ", .1387819026 10 , " g(h)= ", .1476061262 10 "******************" "******************" "h= ", 35, " f(h)= ", 4866375939149411126568541221978609130187, " g(h)= ", 5166247914920387454133563192733079560543 ", which is exactly (for small h) or approximately (for large h)\ : " 40 40 " f(h)= ", .4866375939 10 , " g(h)= ", .5166247915 10 "******************" "******************" "h= ", 36, " f(h)= ", 175495443952717615338214695563848399853498, " g(h)= ", 185985984354936719438428531698824747287399 ", which is exactly (for small h) or approximately (for large h)\ : " 42 42 " f(h)= ", .1754954440 10 , " g(h)= ", .1859859844 10 "******************" "******************" "h= ", 37, " f(h)= ", 6504025044414992880568512925332676964424375, " g(h)= ", 6881516006536896674981607489737909629272187 ", which is exactly (for small h) or approximately (for large h)\ : " 43 43 " f(h)= ", .6504025044 10 , " g(h)= ", .6881516007 10 "******************" "******************" "h= ", 38, " f(h)= ", 247537478597040125902742769348175393825836350, " g(h)= ", 261498772457029239810224392607988837004693085 ", which is exactly (for small h) or approximately (for large h)\ : " 45 45 " f(h)= ", .2475374786 10 , " g(h)= ", .2614987725 10 "******************" "******************" "h= ", 39, " f(h)= ", 9668173858310135478996405747346646383176745603, " g(h)= ", 10198492496495808090446797933825991277918391874 ", which is exactly (for small h) or approximately (for large h)\ : " 46 47 " f(h)= ", .9668173858 10 , " g(h)= ", .1019849250 10 "******************" "******************" "h= ", 40, " f(h)= ", 387266473856059059671608351358161049006040794722, " g(h)= ", 407941140681917348313149106842678580801746168417 ", which is exactly (for small h) or approximately (for large h)\ : " 48 48 " f(h)= ", .3872664739 10 , " g(h)= ", .4079411407 10 "******************" "******************" "h= ", 41, " f(h)= ", 15898946809919081374986269399499189199841647237935, " g(h)= ", 16725639649034946401795461609034264084354511170068 ", which is exactly (for small h) or approximately (for large h)\ : " 50 50 " f(h)= ", .1589894681 10 , " g(h)= ", .1672563965 10 "******************" "******************" "h= ", 42, " f(h)= ", 668595874346481752211759595112118518568443381491750, " g(h)= ", 702478859575409826183701119474805169103048039592560 ", which is exactly (for small h) or approximately (for large h)\ : " 51 51 " f(h)= ", .6685958743 10 , " g(h)= ", .7024788596 10 "******************" "******************" "h= ", 43, " f(h)= ", 28784038214508916526310780336637647029693826076104635, " g(h)= ", 30206668188776216445303196903668695350215399008859142 ", which is exactly (for small h) or approximately (for large h)\ : " 53 53 " f(h)= ", .2878403821 10 , " g(h)= ", .3020666819 10 "******************" "******************" "h= ", 44, " f(h)= ", 1267941995842041336223682855280722654636118199820052362, " g(h)= ", 1329096468779816770897270706138983837486049578163089241 ", which is exactly (for small h) or approximately (for large h)\ : " 55 55 " f(h)= ", .1267941996 10 , " g(h)= ", .1329096469 10 "******************" "******************" "h= ", 45, " f(h)= ", 57119448966149028795781157349279330832412772881052964647, " g(h)= ", 59809466111042766399853760230928754644872011094707857426 ", which is exactly (for small h) or approximately (for large h)\ : " 56 56 " f(h)= ", .5711944897 10 , " g(h)= ", .5980946611 10 "******************" "******************" "h= ", 46, " f(h)= ", 2630223324972771992194913578954625717673170347060174120334, " g(h)= ", 2751240660570446267425929793939981718503214283885055252100 ", which is exactly (for small h) or approximately (for large h)\ : " 58 58 " f(h)= ", .2630223325 10 , " g(h)= ", .2751240661 10 "******************" "******************" "h= ", 47, " f(h)= ", 123743204230298587511749402999242805440843259645707826209907, " g(h)= ", 129308534221321527455809457483970424950648704592046475674867 ", which is exactly (for small h) or approximately (for large h)\ : " 60 60 " f(h)= ", .1237432042 10 , " g(h)= ", .1293085342 10 "******************" "******************" "h= ", 48, " f(h)= ", 5945314776141310424295158037021971962523195783067925373733682 , " g(h)= ", 6206819409950297598721124901512738446018210572124540807321294 ", which is exactly (for small h) or approximately (for large h)\ : " 61 61 " f(h)= ", .5945314776 10 , " g(h)= ", .6206819410 10 "******************" "******************" "h= ", 49, " f(h)= ", 2915853892966334679149950348630546956366786\ 93202607473820232351, " g(h)= ", 304134588392917868317204\ 538639324026326699832541780126009376067 ", which is exactly (for small h) or approximately (for large h)\ : " 63 63 " f(h)= ", .2915853893 10 , " g(h)= ", .3041345884 10 "******************" "******************" "h= ", 50, " f(h)= ", 1459198046857256987337717733888956907894835\ 7175544937832590283126, " g(h)= ", 1520674943892031373571\ 8988921891666957488791414690892747031888674 ", which is exactly (for small h) or approximately (for large h)\ : " 65 65 " f(h)= ", .1459198047 10 , " g(h)= ", .1520674944 10 "******************" "******************" "h= ", 51, " f(h)= ", 7448135009518040808630071502841351001472231\ 43234080799017688134827, " g(h)= ", 775545157982438559516\ 784967165140945666994714447639759391255073883 ", which is exactly (for small h) or approximately (for large h)\ : " 66 66 " f(h)= ", .7448135010 10 , " g(h)= ", .7755451580 10 "******************" "******************" "h= ", 52, " f(h)= ", 3876141058611602130970871930148418613358157\ 7868797708536817764022618, " g(h)= ", 4032839297558093252\ 6258002136839225581932761862308657444918264747355 ", which is exactly (for small h) or approximately (for large h)\ : " 68 68 " f(h)= ", .3876141059 10 , " g(h)= ", .4032839298 10 "******************" "******************" "h= ", 53, " f(h)= ", 2055940501869053053581895346303167415217453\ 177334792454984694163232151, " g(h)= ", 21374070118376619\ 36413473129553184522647622570099140781009892785918028 ", which is exactly (for small h) or approximately (for large h)\ : " 70 70 " f(h)= ", .2055940502 10 , " g(h)= ", .2137407012 10 "******************" "******************" "h= ", 54, " f(h)= ", 1111032061284631556996623384743916176956802\ 31107619446469027646150409182, " g(h)= ", 115420087411473\ 094220894135709208598271253619421429408571551346513449706 ", which is exactly (for small h) or approximately (for large h)\ : " 72 72 " f(h)= ", .1111032061 10 , " g(h)= ", .1154200874 10 "******************" "******************" "h= ", 55, " f(h)= ", 6115042542649566792449398949509307050608048\ 993174108680409297128766662755, " g(h)= ", 63481103339167\ 8400161969819666336231242563873019487839419650569835638023\ 0 ", which is exactly (for small h) or approximately (for large h)\ : " 73 73 " f(h)= ", .6115042543 10 , " g(h)= ", .6348110334 10 "******************" "******************" "h= ", 56, " f(h)= ", 3426780539014101643462316067103498160098881\ 20950776388137625462586580744706, " g(h)= ", 355494465020\ 0196369531086934096234238036408687167999782103499438759275\ 72002 ", which is exactly (for small h) or approximately (for large h)\ : " 75 75 " f(h)= ", .3426780539 10 , " g(h)= ", .3554944650 10 "******************" "******************" "h= ", 57, " f(h)= ", 1954560554374079282534789733502400412130913\ 2635353412411099847903527948656655, " g(h)= ", 2026319962\ 8204203188069961964868875413769483078179347042874838946976\ 049785249 ", which is exactly (for small h) or approximately (for large h)\ : " 77 77 " f(h)= ", .1954560554 10 , " g(h)= ", .2026319963 10 "******************" "******************" "h= ", 58, " f(h)= ", 1134370390452672402322123020153403573055512\ 963608323394906386674449568018914502, " g(h)= ", 11752663\ 9229556210692697808484680749240123788084051883491319063667\ 1901235088963 ", which is exactly (for small h) or approximately (for large h)\ : " 79 79 " f(h)= ", .1134370390 10 , " g(h)= ", .1175266392 10 "******************" "******************" "h= ", 59, " f(h)= ", 6696917730016135317738301980577250109746166\ 0319069438555717162313908344033003419, " g(h)= ", 6934076\ 1763991512731155367313029776957349884297839131994922860227\ 256791860389063 ", which is exactly (for small h) or approximately (for large h)\ : " 80 80 " f(h)= ", .6696917730 10 , " g(h)= ", .6934076176 10 "******************" "******************" "h= ", 60, " f(h)= ", 4020546549770437273152006230270850883653990\ 532143472668109824691000418416797316394, " g(h)= ", 41604\ 4819677083546732834698373079721774665631515624723046127066\ 5588159017843358114 ", which is exactly (for small h) or approximately (for large h)\ : " 82 82 " f(h)= ", .4020546550 10 , " g(h)= ", .4160448197 10 "******************" "******************" "h= ", 61, " f(h)= ", 2453946475092665903302416039641191198099746\ 68898305905150508694461576009713387659783, " g(h)= ", 253\ 7874815650071602827533664969906850020262989530934367208392\ 29287182708503331789968 ", which is exactly (for small h) or approximately (for large h)\ : " 84 84 " f(h)= ", .2453946475 10 , " g(h)= ", .2537874816 10 "******************" "******************" "h= ", 62, " f(h)= ", 1522294368882169778266299005925944961396981\ 9714212767348484068911992813628392396176430, " g(h)= ", 1\ 5734832044194579074681528343604051378978021825319165358294\ 401874410192721682574351965 ", which is exactly (for small h) or approximately (for large h)\ : " 86 86 " f(h)= ", .1522294369 10 , " g(h)= ", .1573483204 10 "******************" "******************" "h= ", 63, " f(h)= ", 9595622880642237508586071033219977277536047\ 70571621951135378743547796967059849461230163, " g(h)= ", 9912949004995634830606896417978192712965637100336446324890\ 64411664942322415083445091693 ", which is exactly (for small h) or approximately (for large h)\ : " 87 87 " f(h)= ", .9595622881 10 , " g(h)= ", .9912949005 10 "******************" "******************" "h= ", 64, " f(h)= ", 6144401993992709218438903915513284098961768\ 3902387587569148076468789950048725833885094610, " g(h)= ", 6344290245835638672534692571110858438092928659230945812180\ 6810260399020207300446248948682 ", which is exactly (for small h) or approximately (for large h)\ : " 89 89 " f(h)= ", .6144401994 10 , " g(h)= ", .6344290246 10 "******************" "******************" "h= ", 65, " f(h)= ", 3995878780446407613755670531808977442734170\ 779133367342493688287796263846615864407939830655, " g(h)= ", 4123790413712726671562360391651682466057143938596555337236\ 833017487552233142688703506455021 ", which is exactly (for small h) or approximately (for large h)\ : " 91 91 " f(h)= ", .3995878780 10 , " g(h)= ", .4123790414 10 "******************" "******************" "h= ", 66, " f(h)= ", 2638570798134135456570270946436147653417540\ 91181044355904377880222530880619065786147415081494, " g(h)= ", 2721702757801318408527020156982463902858439026\ 71352676890285778122209671016574778020254484121 ", which is exactly (for small h) or approximately (for large h)\ : " 93 93 " f(h)= ", .2638570798 10 , " g(h)= ", .2721702758 10 "******************" "******************" "h= ", 67, " f(h)= ", 1768681213808398365545862969879502906649605\ 7474424966225308145980287884330980500320194865055883, " g(h)= ", 1823541529494250125350704908574967257742613911\ 4511800947584969687255996443638209264457701233889 ", which is exactly (for small h) or approximately (for large h)\ : " 95 95 " f(h)= ", .1768681214 10 , " g(h)= ", .1823541529 10 "******************" "******************" "h= ", 68, " f(h)= ", 1203256664630265013620070602861905325756337\ 404554482161776777013275510857796699059892893428366074, " g(h)= ", 1240008675384381508595830783330942303382582621\ 391351720702291284009114875601684562823881831824765 ", which is exactly (for small h) or approximately (for large h)\ : " 97 97 " f(h)= ", .1203256665 10 , " g(h)= ", .1240008675 10 "******************" "******************" "h= ", 69, " f(h)= ", 8306178025075410697786388426784588515290457\ 9354432121638062172990185847152203082851464752769821815, " g(h)= ", 8556062683499297279086680355284685301966360215\ 0649003070032307461386064970147003197227096758662087 ", which is exactly (for small h) or approximately (for large h)\ : " 98 98 " f(h)= ", .8306178025 10 , " g(h)= ", .8556062683 10 "******************" "******************" "h= ", 70, " f(h)= ", 5816844743768464866177932900605648705627495\ 525199119134008047703317371839222628244260579341133665918, " g(h)= ", 5989245737854038001869290338745444007116672120\ 402669477724944081647096875454871231100527965172730974 ", which is exactly (for small h) or approximately (for large h)\ : " 100 100 " f(h)= ", .5816844744 10 , " g(h)= ", .5989245738 10 "******************" "******************" "h= ", 71, " f(h)= ", 4131698214065360911449521145899532540996844\ 3742695673978729533133141182691125422047802123409822843552\ 3, " g(h)= ", 4252365717086355695419796582109823528630532\ 1024087954461034925539205796957100446812213366389184712895\ 1 ", which is exactly (for small h) or approximately (for large h)\ : " 102 102 " f(h)= ", .4131698214 10 , " g(h)= ", .4252365717 10 "******************" "******************" "h= ", 72, " f(h)= ", 2976039327266975162142614584513420097063620\ 7498716178993316735167360401119882638232544605056283439371\ 682, " g(h)= ", 30617041599824364529136572263875606896359\ 2403948961272962268371774722392118409685805425998035001888\ 51944 ", which is exactly (for small h) or approximately (for large h)\ : " 104 104 " f(h)= ", .2976039327 10 , " g(h)= ", .3061704160 10 "******************" "******************" "h= ", 73, " f(h)= ", 2173372298483918861251849617099925975035677\ 0107904042484194603110081335625215834584623595543356940951\ 97423, " g(h)= ", 223504461779073916271744968545384407042\ 8631615657597311224121354593051864682114010598489396274429\ 559452720 ", which is exactly (for small h) or approximately (for large h)\ : " 106 106 " f(h)= ", .2173372298 10 , " g(h)= ", .2235044618 10 "******************" "******************" "h= ", 74, " f(h)= ", 1608917141753171294313365860811710402988839\ 5066010766984987377382733865952113302713360365063194063650\ 3630182, " g(h)= ", 1653933423098478799906642801264419167\ 2486625782657859745039774617669851651136401561548800643058\ 3882915015454 ", which is exactly (for small h) or approximately (for large h)\ : " 108 108 " f(h)= ", .1608917142 10 , " g(h)= ", .1653933423 10 "******************" "******************" "h= ", 75, " f(h)= ", 1207141552459178785404420110834657725438170\ 3851242041275904173043543976151287509888413378280170430014\ 766913403, " g(h)= ", 12404503550079309552278250544990469\ 4652569119359689320245733905517395580704133881073921413075\ 90116892306848311 ", which is exactly (for small h) or approximately (for large h)\ : " 110 110 " f(h)= ", .1207141552 10 , " g(h)= ", .1240450355 10 "******************" "******************" "h= ", 76, " f(h)= ", 9177632419887077930717170459931036208006198\ 5587772416575737086985724818022051367135451528831637508152\ 8547560138, " g(h)= ", 9427424765699805468454570217554375\ 1593715278418764129856698555139874007792661689205376639032\ 6183314648901686658 ", which is exactly (for small h) or approximately (for large h)\ : " 111 111 " f(h)= ", .9177632420 10 , " g(h)= ", .9427424766 10 "******************" "******************" "h= ", 77, " f(h)= ", 7069293897477601646977513035227456085532162\ 2690641292916040764754406197199484165506016037819111543129\ 866390631655, " g(h)= ", 72591185763586500273464124702948\ 7191032433337372902312058560204563418329413328219031841857\ 24910484269570336860880 ", which is exactly (for small h) or approximately (for large h)\ : " 113 113 " f(h)= ", .7069293897 10 , " g(h)= ", .7259118576 10 "******************" "******************" "h= ", 78, " f(h)= ", 5515961717492633746039847305569659306369417\ 2474680629446780938181961270978078575355425752784456047915\ 24012466555598, " g(h)= ", 566211360269917142599753462192\ 7309143110706202926190553880818152593092097128772113369548\ 419416261702472699051216525 ", which is exactly (for small h) or approximately (for large h)\ : " 115 115 " f(h)= ", .5515961717 10 , " g(h)= ", .5662113603 10 "******************" "******************" "h= ", 79, " f(h)= ", 4359082070795491380282652860409039606984529\ 5853776470569900579394806558759587101052607935966445660236\ 5831520994031667, " g(h)= ", 4473070579626484872328823812\ 8995567116697568164852135611009203685315428743437804718337\ 8043181171894557524056207851490 ", which is exactly (for small h) or approximately (for large h)\ : " 117 117 " f(h)= ", .4359082071 10 , " g(h)= ", .4473070580 10 "******************" "******************" "h= ", 80, " f(h)= ", 3488413838874037412991268526422740649481662\ 8445317774078040655437060094840088041771505355317404089273\ 797495525988893298, " g(h)= ", 35784570961534792025795413\ 5270515457563463895588345208319417025398688290619750054296\ 12965769825827513632301363180092853 ", which is exactly (for small h) or approximately (for large h)\ : " 119 119 " f(h)= ", .3488413839 10 , " g(h)= ", .3578457096 10 "******************" "******************" "h= ", 81, " f(h)= ", 2826522102395781504468973771146321289158596\ 5088051668767922050649198467742649064696827157770313588870\ 95035839684452734047, " g(h)= ", 289855073412265507056216\ 6185331326013396968173988919914467340621733802863673344527\ 886629058713996796193579876480400535446 ", which is exactly (for small h) or approximately (for large h)\ : " 121 121 " f(h)= ", .2826522102 10 , " g(h)= ", .2898550734 10 "******************" "******************" "h= ", 82, " f(h)= ", 2318473505160243853403725975430379504476986\ 2046622159497683566190456968668097872923247207109868492428\ 6318334989499875752630, " g(h)= ", 2376811980678385122410\ 0514949398979205404553562625259501925069250237414552016615\ 5853042889836573080007688048322518380889065 ", which is exactly (for small h) or approximately (for large h)\ : " 123 123 " f(h)= ", .2318473505 10 , " g(h)= ", .2376811981 10 "******************" "******************" "h= ", 83, " f(h)= ", 1924920463105599395785663505747049809101750\ 9471346699208535490687741823098533526118579745931103928211\ 051359424743983728194667, " g(h)= ", 19727542426982751073\ 3646310090281302945823828330298284914655231390596790308943\ 79573486406568369784112366141652062513746586537 ", which is exactly (for small h) or approximately (for large h)\ : " 125 125 " f(h)= ", .1924920463 10 , " g(h)= ", .1972754243 10 "******************" "******************" "h= ", 84, " f(h)= ", 1617414817363065800310057177808696173117422\ 7871290247956360214237260160765470168388768667642147676822\ 22031182636347134662337690, " g(h)= ", 165711380251564581\ 9542546003095381174733184099977793806364002421756744612385\ 164096414857775698836854955774409811867389220998368 ", which is exactly (for small h) or approximately (for large h)\ : " 127 127 " f(h)= ", .1617414817 10 , " g(h)= ", .1657113803 10 "******************" "******************" "h= ", 85, " f(h)= ", 1375202278569782889117402508676755014589140\ 3261615518347819531807331515943695800204008363935851367811\ 5434042674773722142774672215, " g(h)= ", 1408546925176993\ 9101062558446919260068105809700926989550085367476862236821\ 9206854929512596650514226798655311694782275121558024126 ", which is exactly (for small h) or approximately (for large h)\ : " 129 129 " f(h)= ", .1375202279 10 , " g(h)= ", .1408546925 10 "******************" "******************" "h= ", 86, " f(h)= ", 1183009638549394307673051058050446657683453\ 3834745747307837428451341543805097478692158545194358370234\ 868162349640275272624800225566, " g(h)= ", 12113505137309\ 0787015868019306972321029458317278262163358666137680024718\ 1968833986262543532219934932773004384610357854747054658301\ 0 ", which is exactly (for small h) or approximately (for large h)\ : " 131 131 " f(h)= ", .1183009639 10 , " g(h)= ", .1211350514 10 "******************" "******************" "h= ", 87, " f(h)= ", 1029503666759629571079894936616360121294246\ 1169287495189674428208658252579636271162430760030704195960\ 67907290674499339939458887500835, " g(h)= ", 105387507797\ 9133363226826920924983701961194808978488870974959177460688\ 9650937438627855994852092561348111703375866604248834365302\ 50450 ", which is exactly (for small h) or approximately (for large h)\ : " 133 133 " f(h)= ", .1029503667 10 , " g(h)= ", .1053875078 10 "******************" "******************" "h= ", 88, " f(h)= ", 9062085301071182865971705347293424484140545\ 6285820042243945968894763977946864602111235884539919012036\ 388681209659917070104397504944450, " g(h)= ", 92741017854\ 8851698425440830793677012462935199546883726546973167934822\ 3810561320514312313915548818387137974003839828241661268931\ 1586475 ", which is exactly (for small h) or approximately (for large h)\ : " 134 134 " f(h)= ", .9062085301 10 , " g(h)= ", .9274101785 10 "******************" "******************" "h= ", 89, " f(h)= ", 8067389730596557297289422013618537425921822\ 2512579551933232202537406247378926689590559812208270439234\ 69194499731763689457152799121875407, " g(h)= ", 825395152\ 2301341152040147471771137378973901756191519727686041098153\ 5984792572813115198467714658359065935932903202260490950485\ 01939981317 ", which is exactly (for small h) or approximately (for large h)\ : " 136 136 " f(h)= ", .8067389731 10 , " g(h)= ", .8253951522 10 "******************" "******************" "h= ", 90, " f(h)= ", 7262528232313170179553253394874283717017459\ 8118217243484015210458939962397545298660084525615316772621\ 9065171054595064712728181475306196486, " g(h)= ", 7428557\ 1716616091490173970520220993600274679563104189930205511706\ 9479036066322584912296432089940063324096703925620743238676\ 819325020379989 ", which is exactly (for small h) or approximately (for large h)\ : " 138 138 " f(h)= ", .7262528232 10 , " g(h)= ", .7428557172 10 "******************" "******************" "h= ", 91, " f(h)= ", 6610571400480765794214898102623873519008328\ 5768472512122505779638557650115556046628254270628214172924\ 375360201316426410106367384666839097179, " g(h)= ", 67599\ 8772276836465510384735077316913883069179579818739589910664\ 8142891795489069912334683097093216570042429516383442113532\ 4541492530610368074 ", which is exactly (for small h) or approximately (for large h)\ : " 140 140 " f(h)= ", .6610571400 10 , " g(h)= ", .6759987723 10 "******************" "******************" "h= ", 92, " f(h)= ", 6083229112720422977585583329930362218561084\ 8054614197405219751770785964718262584133072177245316438800\ 89560462238107839763336438353246822353002, " g(h)= ", 621\ 9189317206688362383095242987703280513371935333348268703541\ 4308126483964959412354969635107782220630800120304589389287\ 36805238135975921205732 ", which is exactly (for small h) or approximately (for large h)\ : " 142 142 " f(h)= ", .6083229113 10 , " g(h)= ", .6219189317 10 "******************" "******************" "h= ", 93, " f(h)= ", 5658771000880139239036086991742964277434685\ 3240470844248702050469115571043483631553586456849435474401\ 5831104441900469170457364202667515285307335, " g(h)= ", 5\ 7838466092974499026196670169469985361791582956512721344005\ 1781975288122503976434715316108964320444432082465280249904\ 761693482502870032359140215 ", which is exactly (for small h) or approximately (for large h)\ : " 144 144 " f(h)= ", .5658771001 10 , " g(h)= ", .5783846609 10 "******************" "******************" "h= ", 94, " f(h)= ", 5320503062034947762733296730226372658017403\ 1838361702661274031292183097371705613040511107225762489619\ 646762029477248318143855110712467703072735598, " g(h)= ", 5436816302064525124725181108974186647698833438394084914587\ 7459065128431773383718330655917663137235984967017463612820\ 502987685097711377301332410111 ", which is exactly (for small h) or approximately (for large h)\ : " 146 146 " f(h)= ", .5320503062 10 , " g(h)= ", .5436816302 10 "******************" "******************" "h= ", 95, " f(h)= ", 5055647992514661027838708793302016587345984\ 5136823018921146190772742532486665561456620063263577235375\ 56179868853035267912291826649400225574108380691, " g(h)= ", 5164975931766791900087918749790150547361378016091913671280\ 7737012631607940001139231553007878300582494044116388768947\ 22322052481762354196227299603534 ", which is exactly (for small h) or approximately (for large h)\ : " 148 148 " f(h)= ", .5055647993 10 , " g(h)= ", .5164975932 10 "******************" "******************" "h= ", 96, " f(h)= ", 4854521808873516294644266474480407058938969\ 9990599940256753696416640762462822890243859418755820878747\ 4138759717254567654219765518661801309198141725202, " g(h)= ", 4958377303285981232452504431144163879393226241\ 8149345270286971220878756167780095805450007214417215813181\ 9681504452317212997006909214633624989217743511 ", which is exactly (for small h) or approximately (for large h)\ : " 150 150 " f(h)= ", .4854521809 10 , " g(h)= ", .4958377303 10 "******************" "******************" "h= ", 97, " f(h)= ", 4709930771534691238288602125332203702579836\ 5458709300961294780437130229297879006506931282661727197669\ 662331759361716596317959271409886245773573948362047, " g(h)= ", 4809626363970451890466507245257400464870882558\ 2807902715510544536924963811014466510103128418314487672687\ 268245031996718650256232990013132518293221792159 ", which is exactly (for small h) or approximately (for large h)\ : " 152 152 " f(h)= ", .4709930772 10 , " g(h)= ", .4809626364 10 "******************" "******************" "h= ", 98, " f(h)= ", 4616734864150799350136112711206608210851731\ 5178684376373695332463111552883529006456607973673767677422\ 52301972062165178954680100535907671650185618220292950, " g(h)= ", 4713434193327686354741975681730044802757109089\ 1823456779847698722402464082794369087427045936079293997033\ 99384844185037092149358127772503092295581306933303 ", which is exactly (for small h) or approximately (for large h)\ : " 154 154 " f(h)= ", .4616734864 10 , " g(h)= ", .4713434193 10 "******************" "******************" "h= ", 99, " f(h)= ", 4571540024493796408292367403401530463955367\ 0671941061481619750629396797441714680765244180081849602753\ 2747737887567519695020248579134183801103924226095011915, " g(h)= ", 4666300189865660073520565894440634865811501941\ 4333012439302347644871536485998415667791639833628309957162\ 8728139634768834218786447953097037684049700217802171 ", which is exactly (for small h) or approximately (for large h)\ : " 156 156 " f(h)= ", .4571540024 10 , " g(h)= ", .4666300190 10 "******************" "******************" "h= ", 100, " f(h)= ", 457249297035222784980048694393983212722767\ 3545202731481651244432820191966459221532950837958089688835\ 4476246426632210117889758785953718691201692612066970613306, " g(h)= ", 4666300514485158296402274322204901463839367594\ 2294818488060200326708844394572102663679223692209862830282\ 250013360549818627829410391422578476758494039360841845 ", which is exactly (for small h) or approximately (for large h)\ : " 158 158 " f(h)= ", .4572492970 10 , " g(h)= ", .4666300514 10 "******************" "******************" >