Certain classses of numbers are inherently carréphylic. Here are the five that come with every square and two more that come with even squares:

They show convincingly that members of the same class have a similar approach profile.


Carréphylic classes: n2-2

Here are the first 10 carréphylic numbers of the form n2-2 with the usual data.
Note that all fractions except the s/p- and t/q-fractions are below the root-value.
A nice tq-pattern over even and odd squares.
For all t/q fractions involved: v = 2  
For q/t-fractions 'q' follows: U(1,7,4)2 1, 7, 17, 31, 49, 71, ...
while 'v' follows its negative: U(-1,-7,-4)2 -1, -7, -17, -31, -49, -71, ...

√7
1012358212937458212733646359071713072024535573799403114272083032257853441176011498581821153319735140881360149187423723883252902413529073881931512167704029870191380633424625649384319835130576328345472491...
01111238111417314812717522327149476520242789355443197873121923225744449566416883312547419430751408870839590270210970091999711309672081931511128987114386591174833113186990249353213130576328...

√14
101234711155671861011162173334491680212925783027347665039979134555034463799772549070910416419487329903740320115086401911841231504227182433121444583968789611311208257545208856...
011111234151923273158891204495696898099291738266735961345517051206472424327839520827992110776040320151096161872172648183424115607222394963322920412082575...

√23
101234514192411513916318721123568191611515520667178228973101241127532674439495522426484532006937529343051748574154096515676712108636264960112707040153566411800624220655843233054442595504575215534101170579127125624609673075...
01111113452429343944491421912401151139116311871211123516813916411515552246673937529389769101284112799326882439681552480264960132020813754561430704148595215412001156835232109552426507525127125624...

√34
1012345617232935204239274309344379414120716212035244914280167291917821627240762652528974844731134471424211713959993961170791134218615135811684976185637120277665911903793966999674351199520169943440...
01111111345635414753596571207278349420244928693289370941294549496914487194562442529394171395200789230183259577288971318365347759101388313616421709401205716011995201...

√47
10123456727344148329377425473521569617665261232773942460731584361914079845405500125461959226638332507253145583783914422243031735347395939161834358407480063152428555685079612730324066988301942913632159442448897291014976...
011111111456748556269768390973814785756724607527959516623729579678639931136572458835519464505442224506729571234635739700244764749829254893759351053144042905298049619180842448897...

√62
1012345678313947556349655962268574881187493710003937493759376937793762496704337837086307942441021811101181180551259924906316220237480158740079999997874000...
01111111114567863717987951031111191275006277548811008793789459953109611196912977139851499316001629967899794998110999127000999999...

√79
1012345678944536271807117918719511031111111911271135114317075850699371136812799113760...
011111111115678980899810711612513414315216179695711181279144012799...

√98
1012345678910495969798999980107911781277137614751574167317721871197097511172113691156611763119601194040...
0111111111115678910991091191291391491591691791891999851184138315821781198019601...

√119
10123456789101165768798109120130914291549166917891909202921492269238925092629156541828320912235412617028799314160...
01111111111116789101112013114215316417518619720821923024114351676191721582399264028799...

√142
10123456789101112718395107119131143170418471990213322762419256227052848299131343277342020377237972721730637340573747740897487344...
011111111111116789101112143155167179191203215227239251263275287171019972284257128583145343240897...



Carréphylic classes: n2-1

Here are the first 10 carréphylic numbers of the form n2-1 with the usual data.
For numbers 'n' that are one less than a square, or N2-n = 1, the formula gives the sp-fraction and factor of the base-2 accelleration of the series.
The first fraction is N/1, F=2N, because obviously N2-n*12 = 1 is a solution of the diophantine equation.
Note that the q/t fractions immediately precede the sp-blocks, that their v-value decreases with steps of 2 over 'n' and that all approximations except the s/p-fractions are below the root-value.
Compare n2+n.

√3
10123571219264571971682653626279891351234036915042873313775188173259251409702261216351918612620874539487160359781221694157...
01112347111526415697153209362571780135121312911504279531086418817296814054570226110771151316262087413403564719978122...

√8
101238111417486582992803794785771632220927863363951212875162381960155440750419464211424332312843737155161466585718833282549185321504238808991097684014857739187386382261953763977712...
0111134561723293599134169204577781985118933634552574169301960126531334614039111424315463419502523541666585790127311366891372105388089952530046625109799721422619537...

√15
10123415192327311201511822132449451189143316771921744093611128213203151245857573699888231039471190714611605802316993028183739374443630705456814955055936443037738048128584480...
01111145678313947556324430737043349619212417291334093905151241902922934268393074411907114981518055921130324204793744411794911421538166358519056327380481...

√24
1012345242934394449240289338387436485237628613346383143164801235202832133122379234272447525232824280349327874375399422924470449230472027751693245618371606741865164656965228143762747134132128306367852714144223646099201225839040...
0111111567891049596979899948558468378288198048015781676177418721970147525572266692776628863299603047044956647966250975853985456995059946569655607564655816375087628459361940996046099201...

√35
10123456354147535965714204915626337047758465005585166977543838992351008159640697217980289883999641100451201267106758308019509271071053119117913113051431431846846098998911133132212762753141941841562561517057046100910845...
0111111167891011127183951071191311438469891132127514181561170410081117851348915193168971860120305120126140431160736181041201346221651241956143143116733871915343215729923992552641211288316717057046...

√48
101234567485562697683909767276986696310601157125413519360107111206213413147641611517466188171303681491851680021868192056362244532432702620871815792207787923399662602053286414031262273388314365040125290720...
0111111117891011121314971111251391531671811951351154617411936213123262521271618817215332424926965296813239735113378292620872999163377453755744134034512324890615268903650401...

√63
10123456786371798795103111119127100811351262138915161643177018972024160651808920113221372416126185282093023332257256032...
0111111111891011121314151612714315917519120722323925520242279253427893044329935543809406432257...

√80
101234567898089981071161251341431521611440160117621923208422452406256727282889258402872931618345073739640285431744606348952518414636805155215673626192036710447228857747268265678784089302498320400925064910180898111111471204139612971645139018941483214315762392166926411493035201659961611826888021993814432160740842327667252494593662661520072828446482995372892679142960...
011111111119101112131415161718161179197215233251269287305323288932123535385841814504482751505473579651841576376343369229750258082186617924139820910400593024910342541138259124226413462691450274155427916582841762289186629416692641185589352042522922291523241578172602411127890405297566993162299333489287299537289...

√99
1012345678910991091191291391491591691791891991980217923782577277629753174337335723771397039501...
01111111111110111213141516171819201992192392592792993193393593793993970...

√120
1012345678910111201311421531641751861972082192302412640...
0111111111111111213141516171819202122241...



Carréphylic classes: n2+1

Here are the first 10 carréphylic numbers of the form n2+1 with the usual data. The qt-blocks in the middle split the sections in a left part below and right part above the root-value. For all q/t fractions v=-1.
A fraction for which v=-1 is always a q/t fraction, always in the middle, and constitutes the only occasion where the denominator of the next fraction in a section surpasses the numerator of the previous one, in carréphylic as well as carréphocic numbers.
To understand the nature of the exception one would have to understand the nature of the rule, which is that the next denominator at most equals the previous numerator, and that only in the sp-blocks. The explanations of both rule and exception have till now eluded me.

√2
1013471724419914023957781613933363475681191960127720473211142431615642758076658579416641607521388089954884209369319226195373198885654608393131836323186444716...
011235121729709916940857798523783363574113860196013346180782114243195025470832665857113668927442103880899662510915994428226195373861396593222358131836323...

√5
10127920293812316136052168222072889646093491223839603518411159201677612196027106479302492080100301034939405981275204316692641373258805401852170711162228826127299537289669785740...
011134913175572161233305987129228894181547317711231845184175025982093178114160209302491346269176228957028877465176166926412415781731622993102334155133957148299537289...

√10
1012313161960799811748760472122803001372244431849322936273798658011395914133816871770224787096410396813287760432744153671226406803266668933307369639480499124848300...
0111145619253137154191228721949117714055848725386582737936037446955335322207027542332877610396811368457169723320260098432812104588211248483039480499...

√17
1012342125293313616920223526813731641190921778976111531333015507176849059710828112596514364959228073592987957810232271166876597802971449058311781947865739081504...
01111156783341495765333398463528217727053233376142892197326262305513484014364917848921332924816928300914498851732894201590322989129478657...

√26
10123453136414651260311362413464515314136564171468652012652031721369224212347324525253203513728764254014779265304512704780323523137656824296133482658453570353267266138029696433867314874376654100801275861040...
011111167891051617181911016167178189191020520162217241826192811030162826731278342893729104030530451634481738511842541946571105060164076367458237850883895594391061004054100801...

√37
101234564349556167734445175906637368098826247712980118893977510657648247548186138967951074521181091287669120191040785116955112983171427083155584994638601101970912575558141314071568725617243105187989541331485271519474811707464351895453892083443432271432971381658736...
01111111789101112738597109121133145102711721317146216071752106571240914161159131766519417211691499351711041922732134422346112557801555849181162920674092323189257896928347493090529218894832498001228070541311610703425159937342128227143297...

√50
10123456757647178859299700799898997109611951294139311243126361402915422168151820819601138600158201177802197403217004236605256206275807222605725018642777671305347833292853605092388089927442100313229993520389839084797429656964684659550727494546083934407480434953564365499648296045732226591816157137900087683984015433397200...
0111111118910111213149911312714115516918319715901787198421812378257527721960122373251452791730689334613623339005314812353817392822431827470832509837548842388089944297414978583552742560762676625109717395177227936233118670053979777767728549956593222358100945151108667944768398401...

√65
1012345678738189971051131211291040116912981427155616851814194320721877720849229212499327065291373120933281268320...
01111111119101112131415161291451611771932092252412572329258628433100335736143871412833281...

√82
10123456789911001091181271361451541631476163918021965212822912454261727802943295933253635479384224136544308472515019453137481176...
0111111111110111213141516171816318119921723525327128930732532683593391842434568489352185543586853137...

√101
10123456789101111211311411511611711811912012020222124222623282430253226342736283829403044531485615259156621606516468168711727417677180801812040...
01111111111111121314151617181920201221241261281301321341361381401443148325233563460356436683772387639804080801...



Carréphylic classes: n2+2

Nice and orderly. For all q/t-fractions, v = -2, placing the it to the right of the middle of the section but keeping up the 'v = -1 division' of fractions to the left and right being below and above the root-value respectively.

√6
1012512172249120169218485118816732158480111760165612136247525116412163937211462470449115236016228092093258465696511407188...
011125792049698919848568388119604801676187211940247525669278632919206047044966250985456919011984656965...

√11
101237103343536313619966085910581257271339701316717137211072507754124792012626803418814210825002831079767158005052404336820483840053399805832154121631521799104545980...
01111231013161941601992593193798181197397051676364756116319238807920110308112696115084132556247640315800502056453253285630092596494921950418031521799...

√18
101234131772891061231404375772448302536024179475614845196018316010276112236214196316156450429366585728249923490849415670648225635488420171311172261953795966568...
0111113417212529331031365777138499851121349946201960124221288413346138081118863156944665857822801979745113668912936334037843533147622619537...

√27
101234516212613516118721323926582110861351702083719722110731242413775426765645170226364905435131505357575583645809716035221833129343663650401189680402261844126268842299192433356964437220045115310536152530581189750626985973175...
011111134526313641465115820926013511611187121312391265182131086413515702268374197256110771124286137801426918564719702520365040143529215055441575796164604817163001221915232935452436517525189750626...

√38
1012345625313722826530233937641345018372287273716872196092234625083278203055733294135913169207202501124830014508011653302185580320583042260805246330610055725125190311498233717985374410733966512232200213730433915228667616726901318225135074398773792623908711084904376833193972...
01111111456374349556167732983714442737318136254069451349575401220482744932850202501235351268201301051333901366751399601163125420308552430456149823371741279319843249222737052470416127134617295650731206907481502558211798208941108490437...

√51
10123456729364350357407457507557607657707287835854292499935700406994569850697556966069565694706932877713584644291574998503569643406949345693435069193556904360688936568743706859328774222358428154291140849980001356928600...
011111111456750576471788592994035026017004999569963997099779984999199989940296501956009469993499850569843639836709829779822849815919808989801402919750189986008799699860049980001...

√66
1012345678414957655285936587237888539189831048530563537401844968640...
01111111115678657381899710511312112965378291110408449...

√83
10123456789465564738274782991199310751157123913211403148575078992104771196213447122508...
0111111111156789829110010911812713614515416382498711501313147613447...

√102
101234567891061718191101102011211222132314241525162617271828192920301228114311163411837120401206040...
0111111111116789101011111211311411511611711811912011216141716181819202020401...

√123
101234567891011677889100111122135314751597171918411963208522072329245125732695162921898721682243772707229767330132...
01111111111116789101112213314415516617718819921022123224314691712195521982441268429767...



Carréphylic classes: n2+n

Note that the q/t fractions immediately precede the sp-blocks, that their v-value decreases with steps of 1 over 'n' and that all approximations except the s/p-fractions are below the root-value.
Compare n2-1.

√2
1013471724419914023957781613933363475681191960127720473211142431615642758076658579416641607521388089954884209369319226195373198885654608393131836323186444716...
011235121729709916940857798523783363574113860196013346180782114243195025470832665857113668927442103880899662510915994428226195373861396593222358131836323...

√6
1012512172249120169218485118816732158480111760165612136247525116412163937211462470449115236016228092093258465696511407188...
011125792049698919848568388119604801676187211940247525669278632919206047044966250985456919011984656965...

√12
10123724313845973364335306271351468060317382873318817651848400110281812163526208790789611699831432070169415736504011264536016295761199461622359656350843527176127144...
0111127911132897125153181390135117412131252154321881724249296813511375658262087337745413403489061105378036504014704181575796168117411467726250843527...

√20
1012349404958677616172088110421203136428891292015809186982158724476518412318402836813355223873634392049302494160200509044960206986950947788119616692641746517609134440110803704212472968314142232429953728913395714801639108769193864605822381833472537720636537497856124037634880...
011111291113151736161197233269305646288935354181482754731159251841634337502586617982092080109302491138259134626915542791762289373258816692641204252292415781727890405316229936697857429953728936651586343349443750047301156745158512018817445374978561...

√30
10123451160718293104115241132015611802204322842525529128980342713956244853501445543511616163624075240186856298472311008841217045255025113968300165185511906880221619053241693042671955555989361306666360...
01111112111315171921442412853293734174619665291625772238189915510121212081161611373691585771797852009932222014656102550251301586134814713947081441269148783011022221255989361...

√42
101234561384971101231361491623372184252128583195353238694206874956700654497419882947916961004451091942271371472016169915319262902153427238056426077012834838589681338215716441125295000934255906155618029686769978173596594153090001992136600...
0111111121315171921232552337389441493545597649135087491009911449127991414915499168493504822713726218529723333228136732940237743742590989858968136806711771660986265079536405104463031135620123622300153090001...

√56
1012345671511212714215717218720221744933603809425847075156560560546503134551006881141431275981410531545081679631814181948734032013017280342048138236824226883463008450332855436486583968712082575904177121025002871145828621266654371387480121508305871629131621749957373620740492709514080...
011111111215171921232527296044950956962968974980986917981345515253170511884920647224452424326041538804032014570815109615648416187216726017264817803611614602120825751369717715311779169263811854098320155585217701872338478948384180362074049...

√72
101234567817144161178195212229246263280577489654736050662772047781835889359512196011663201859212055222251232447242643252839263035273231286658575649984...
011111111121719212325272931336857764571378184991798510531121231019601219112422126531288413115133461357713808178472665857...

√90
10123456789191801992182372562752943133323517216840...
0111111111121921232527293133353776721...

√110
1012345678910212202412622833043253463673884094308819240...
0111111111112212325272931333537394184881...



Carréphylic classes: n2-n/2 (for even n)

Here are the first 6 carréphylic numbers of the form n2-n/2 (for even n) with the usual data.
Root-3 is a 'small numbers exception' on the general pattern because it is also a member of n2-1 and shows the characteristic 'sp-block shift' of that class.
The first non-trivial ab-blocks (second for root-3) follow a simple pattern (4n-1)/4. The s/p- and t/q-fractions are the only ones above the root-value.
The values 'v' equal half the numerator of the first t/q fraction and the numerator of the first q/t fraction respectively.

√3
10123571219264571971682653626279891351234036915042873313775188173259251409702261216351918612620874539487160359781221694157...
01112347111526415697153209362571780135121312911504279531086418817296814054570226110771151316262087413403564719978122...

√14
101234711155671861011162173334491680212925783027347665039979134555034463799772549070910416419487329903740320115086401911841231504227182433121444583968789611311208257545208856...
011111234151923273158891204495696898099291738266735961345517051206472424327839520827992110776040320151096161872172648183424115607222394963322920412082575...

√33
1012345611172313215517820122424727051778710576072712981869243103001135712414237713618548599279180327779376378424977473576522175570774109294916637232234497128362081507070517305202195396992177419624008693262431905025188376495073102738263590186388...
011111112342327313539434790137184105712411425160917931977216141386299846048599570596551973979824399089999359190258289617388976223449726234733012449340142537904014179377456835387477301331608317884436102738263...

√60
1012345678152331240271302333364395426457488945143319211488016801187222064322564244852640628327302485857588823119071922320104139111604621279533139860415176751636746175581718748883630705550559373804815716896064549441719299227931040386690884940713651014518461088323271162128082250451353412579434574707513543553200...
011111111123431353943475155596312218524819212169241726652913316134093657390575621146715372119071134443149815165187180559195931211303226675242047468659710769952816738048183332979286113102389291119174512144561130973771405019315003009290453914405621159059220457470751...

√95
101234567891019293938041945849753657561465369273177015012271304129640...
01111111111123439434751555963677175791542333123041...

√138
1012345678910111223354755259964669374078783488192897510221069111621853301441751888...
01111111111111234475155596367717579838791951862813764417...



Carréphylic classes: n2+n/2 (for even n)

Here are the first 6 carréphylic numbers of the form n2+n/2 (for even n) with the usual data.
The first non-trivial sp-blocks follow a simple pattern (4n+1)/n. The qt-blocks follow an even simpler one: n/1.
Their v-value equals -n/2.

√5
10127920293812316136052168222072889646093491223839603518411159201677612196027106479302492080100301034939405981275204316692641373258805401852170711162228826127299537289669785740...
011134913175572161233305987129228894181547317711231845184175025982093178114160209302491346269176228957028877465176166926412415781731622993102334155133957148299537289...

√18
101234131772891061231404375772448302536024179475614845196018316010276112236214196316156450429366585728249923490849415670648225635488420171311172261953795966568...
0111113417212529331031365777138499851121349946201960124221288413346138081118863156944665857822801979745113668912936334037843533147622619537...

√39
101234561925156181206231256281306943124978009049102981154712796140451529447131624253898444522695146945771196395447019697643942355607312000119484400226044012572440228844403319644043508440538204406117733219155937625973830156...
0111111134252933374145491512001249144916491849204922492449754799966242572421824179241310240911240512240137719949960031200013619601411920146188015118401561800161176011885240324970004155937625...

√68
10123456782533272305338371404437470503536164121771795220129223062448326660288373101433191353681082811436491184560...
0111111111343337414549535761651992642177244127052969323334973761402542891313117420143649...

√105
101234567891031414204615025435846256667077487898302531336134440...
0111111111113441454953576165697377812473283361...

√150
10123456789101112374960064969874779684589494399210411090113911883613480158800...
0111111111111134495357616569737781858993972953924801...