Why does the FindSequenceFunction command sometimes succeed with subsequences, while failing for the longer sequences?

Question

I've encountered this phenomenon a number of times--and can provide specific examples if requested. For example, I have a 75-long sequence of rational numbers for which the FindSequenceFunction fails to provide a rule. Contrastingly, if I take the first 50 members of the sequence, the command does furnish a rule, which succeeds in reproducing the last 25 members of the sequence. So, to be totally thorough, it seems that if the command fails, one should undertake the time-consuming activity of checking subsequences of increasing/decreasing length. Only if the command fails for all such subsequences, then one can accept the conclusion that there is no apparent rule. Is there some specific length cut-off rule that Mathematica applies? Here--as suggested by Bob Hanlon--is the 75-long sequence in question. So, if his timing cut-off explanation is correct, is there any way to modify the timing cut-off rule?

{8/33, 26/323, 2999/103385, 44482/4091349, 89514/21460999, 179808469/
  110638410169, 191151001/298529164591, 1331199762/5232880523393, 
  74195568677/729345064647247, 730710456538/17868447453498669, 
  1763088530160992/106791887427691356141, 1884480400168307/
  281722067682582271371, 73311677645345281/26967836824137636644415, 
  456910178894124999/412488768387191176401281, 235858699169080988497/
  521381168536146869542270855, 14201351814342435842/
  76716816738683278836116705, 2178302469068100802897/
  28705960875463688094521233705, 429806807892246603906497/
  13795565957979731569010551915445, 5077791241958454582179372/
  396408873146550339863515593534955, 16915600222887935686875648022/
  3207808012208655726115716096231777847, 
  220072366758124168202760252682/
  101260648196413392173211856927505491249, 36478744649401772516501588/
  40683265647413978374130918813782841, 100873536200588992843564356904/
  272420212154638167052379553997029381313, 
  50868593297571410174924223701080373/
  332366700852321443789422937622477328443605861, 
  234749641791186164106838232918504/
  3707899991729973882890135443497124736181389, 
  3658766751525682214931857205460217/
  139601253828763538992124780933194295003619047, 
  7751132431571389077664649208803390201/
  713922711163771069411457951075078375491144558711, 
  1036767506025579644642168308123737642479/
  230366844136199925641647569249414039437097176944953, 
  5802297075903128151479030066963998381/
  3108359450458298884771594841885865337640896355611, 
  4986795098594104248611638362169017083/
  6437284917664140259622818577919562534636600597933, 
  1779262884821633715492819208764929946535919/
  5531513868754372076112070644276234649075488437599641369, 
  6410605583723568608339707829501538109888638/
  47974716561860127743728451233776222666394343348630654863, 
  67486972876304998119958347845372405417993/
  1215184897735558677895119048928392957368813176164840453, 
  10108651897613196111139625117311363976775529/
  437759883677357983033458920832928318125241077634278903879, 
  16893572462161452090438435924483379095551365184/
  1758759461378453017701928103447444676632452566697670837323269, 
  69792039834426846192333238347065829501936768075643/
  17460826149566003063425383073801536710162275940729158743893642763, 
  6562665684060256928064005413426417965527523830579/
  3944153906465757660242663247744319687717127133714341757940980239, 
  423621636136936099414536534786122915329220927786698/
  611385889653052950000286647453988049049302759674746263659389137727, 
  112836649766195893712883861507429097874360915882434628/
  39093798086583433417035496456749921658243978650498464199064982601945\
3, 2469104765661818826357795748344919918838020451690302994/
  20529643241342147284526501963415880186517451121589728310712410401284\
021, 273234233202269238895216741538172875078179325648406919116/
  54504401959865037694164834072346055239047120670704799741405422210987\
91049, 237572695485036827027842869079327170508804062464640282842/
  11366429831201287176768277117675023110067354367919013524045273000101\
737173, 1513406429160163183067630905213301710465861979451875132522/
  17361865955331721091219496755820213289178178776707676488873738221114\
4917359, 24015844699643455713678450999840521048179907728634703151756/
  66044927233362835042358224531190652061236991577507146302592400985331\
00641589, 
  234295135631403778896477870508995924739356061570136078973839888/
   1544180647001996059640078829426028394685339052457332972114925026354\
28401448489813, 
  20283311909093538693494220127587254955876140039250871626279004184/
   3203052308845810868297928003416351008603116267356943274364188314183\
5394813222994749, 
  7674162014901917726537932116865512909291406576551328397046176856/
   2903005726942520074922753127299384376977622417179875137701458116859\
6049368987123841, 
  94690165605482531317925879428708453600918510467044846981979819741/
   8578652648182939165449633595837219514004029975129682670191812458100\
04058612665408541, 
  74228199976863811380755765181103511603220418505530775649561987369792\
2/16102368177931016000604389365298154821226053794522504987750812030546\
303860462622639468467, 
  15644209801006165966614379673923067197820002213040052573521720666049\
4/81244506782483083764424573238460862040361990328278361869440396392746\
26518069614456111997, 
  54508615141031158027287162454904665119075334148200203336974401449250\
459/677550076156312589151606847313466421029568706768017073431505419128\
5227373100355636335460117, 
  57298151398852165521498313842152765925302311161031467997423588454466\
11/1704402855350628845479941739022307947335817564963935165047026822493\
670450726897758102240653, 
  15448341828485371723604907763733894798396999714411747749815784481035\
73827/1099490388325602979491260965419207631149978994675615122100825413\
287691329591468464503474756637, 
  14385499644470087221892059537063202666733970301905272321850948727986\
7076929/24492765297582696960784736917897289995088061487273849808210740\
2360028650891934769122038759022607, 
  21147305933613335975703724335208049162725414743261318438707033670322\
91239/8611906291558613638586649800147726185841209509249081342862553775\
852392623120673568795731901097, 
  14615656974740672346218082529732663691850335975882207460482126160194\
74979/1423392226824376200799039744416107859202834924822474275363336039\
5103212733169777975724696346917, 
  22818336988249035656534678070253095937092355453761495618797584340543\
88759353027/
   5313556211244822874230576764031644946866140195246164289385437274534\
4155017675385551493728176913537421, 
  45475190386894703927321163053745931297738574246576218449276061029877\
90196453/2531664726627963632966188114948646304667731511735495235440408\
00591757243381243218278753467363146339, 
  84054369879675844974934677741726105090465425461757594304975213092696\
71908251/1118561448341265560270437703543079199403408482266929412518364\
367217988536837236035739912572518720573, 
  21955654046493313002896634958013006674522479333545761029769242176754\
63818765245177/
   6983182535790998199080637074428672543906480474479760371415725881623\
65588828704657277528396814639198437631, 
  44067457226898522027770363554464370585269408680749430046819954134674\
929499784597093/
   3349454010286892133471837995127191356686260920770224027739264923548\
2619782429667798518616809613334632418579, 
  21841232679277943202253146599100069364571950358205265514000908200309\
5103043544620061603/
   3966665740270629934397159248023399730237078388240978321287163685790\
61112288738688196578104436637440362534482309, 
  11409689564245096955310683194320211571600968805189164734264780598727\
28020777433652321/
   4950623795181764306378355978019101062489065090766076039946122490998\
318028074236870721119332804275135560701015, 
  36771078890632459718859906629883387779485082829466124036783284420048\
77441571129859942/
   3811334026755339486628517085323008196567952304603877757541811168918\
4440045899986519603896774691085286460268963, 
  15056452643437982632345115687321173329936655145929814230187482071440\
949676018726905704748031/
   3727574861776103173507020486272410642538946581966828247371570741199\
42155221611443648253609018634666134859559598131515, 
  45169892794341895770842483694396664502862499203348008500974578764958\
32638007432252809348731617471/
   2670766516590101068963588047280235767435762349318095659384398424866\
77485933660721100386303184370483583607516659459390027765, 
  10210118960170926190310107165158089077772959997803727125748756572211\
0524363193945756326384093576/
   1441625237416103143169952199923608880982540854631519738916363400438\
9075139689996288179487526080974538219020487370062803365, 
  43304283224172493326307799406512391195721629930415141316608966694148\
29642158974354016914226534561/
   1459957173364663992429955081754600137724263717999705834551705732099\
078640635851477843215444792295999128225902084026535792835, 
  14508936359063149080780883537181149078655785995285797377380426722104\
434728087984279663716452484482943/
   1167849437018598754608973882541235252202338024578161301244374873527\
7228164135796822415916251119635147060399985868857214356539945, 
  48304748422100741249066840498803221406951302310119608885288328066800\
73258461411018544275580710162/
   9281939867573373170833062451163622376474231205724421024748516560104\
464246463067900060156000923887593568376680246564784680111, 
  23520779384511743636145308774132144300972249140737148050324180131869\
583324431800097056898701352136/
   1078835305919593701331252668504099715560693430304690902712573482149\
84674602333035100699190240246496784261951906472367743249159, 
  87323410789451382253920173940307489286585212883680237113479746396109\
0922272781353100772178001769/
   9559753864203405281725685140123295090223170379293374052586137654154\
105141910597308264589621873811466546673768385105508807121, 
  79035502431495289706996362052889812583373709654824426894793032502308\
244673162271250920957647380/
   2064952280660357226843985271923235276990862097379305616390053107230\
870883877093466309306721333740993219599434371389037804133, 
  90445793185153797800852899676114761492502755265500783536135548655294\
5476658021761049529147712809897/
   5639079985374152293770378927726325333457666729510004720504670634482\
3255593534225439032499478751226896469319518453287523933405033, 
  23553060716641396843799712649572408615393589819050441982985896275120\
20426151676970407951054161695929167/
   3503972336613451404073290700466364645648155714253314102459423921321\
48806886447027343007366165332219929193819666595827008264726888405};

Answer 1

seq = {8/33, 26/323, 2999/103385, 44482/4091349, 89514/21460999, 
   179808469/110638410169, 191151001/298529164591, 1331199762/5232880523393, 
   74195568677/729345064647247, 730710456538/17868447453498669, 
   1763088530160992/106791887427691356141, 
   1884480400168307/281722067682582271371, 
   73311677645345281/26967836824137636644415, 
   456910178894124999/412488768387191176401281, 
   235858699169080988497/521381168536146869542270855, 
   14201351814342435842/76716816738683278836116705, 
   2178302469068100802897/28705960875463688094521233705, 
   429806807892246603906497/13795565957979731569010551915445, 
   5077791241958454582179372/396408873146550339863515593534955, 
   16915600222887935686875648022/3207808012208655726115716096231777847, 
   220072366758124168202760252682/101260648196413392173211856927505491249, 
   36478744649401772516501588/40683265647413978374130918813782841, 
   100873536200588992843564356904/272420212154638167052379553997029381313, 
   50868593297571410174924223701080373/
    332366700852321443789422937622477328443605861, 
   234749641791186164106838232918504/
    3707899991729973882890135443497124736181389, 
   3658766751525682214931857205460217/
    139601253828763538992124780933194295003619047, 
   7751132431571389077664649208803390201/
    713922711163771069411457951075078375491144558711, 
   1036767506025579644642168308123737642479/
    230366844136199925641647569249414039437097176944953, 
   5802297075903128151479030066963998381/
    3108359450458298884771594841885865337640896355611, 
   4986795098594104248611638362169017083/
    6437284917664140259622818577919562534636600597933, 
   1779262884821633715492819208764929946535919/
    5531513868754372076112070644276234649075488437599641369, 
   6410605583723568608339707829501538109888638/
    47974716561860127743728451233776222666394343348630654863, 
   67486972876304998119958347845372405417993/
    1215184897735558677895119048928392957368813176164840453, 
   10108651897613196111139625117311363976775529/
    437759883677357983033458920832928318125241077634278903879, 
   16893572462161452090438435924483379095551365184/
    1758759461378453017701928103447444676632452566697670837323269, 
   69792039834426846192333238347065829501936768075643/
    17460826149566003063425383073801536710162275940729158743893642763, 
   6562665684060256928064005413426417965527523830579/
    3944153906465757660242663247744319687717127133714341757940980239, 
   423621636136936099414536534786122915329220927786698/
    611385889653052950000286647453988049049302759674746263659389137727, 
   112836649766195893712883861507429097874360915882434628/
     39093798086583433417035496456749921658243978650498464199064982601945 3, 
   2469104765661818826357795748344919918838020451690302994/
     20529643241342147284526501963415880186517451121589728310712410401284 021,
    273234233202269238895216741538172875078179325648406919116/
     54504401959865037694164834072346055239047120670704799741405422210987 \
91049, 237572695485036827027842869079327170508804062464640282842/
     11366429831201287176768277117675023110067354367919013524045273000101 \
737173, 1513406429160163183067630905213301710465861979451875132522/
     17361865955331721091219496755820213289178178776707676488873738221114 \
4917359, 24015844699643455713678450999840521048179907728634703151756/
     66044927233362835042358224531190652061236991577507146302592400985331 \
00641589, 234295135631403778896477870508995924739356061570136078973839888/
     1544180647001996059640078829426028394685339052457332972114925026354 \
28401448489813, 
   20283311909093538693494220127587254955876140039250871626279004184/
     3203052308845810868297928003416351008603116267356943274364188314183 \
5394813222994749, 
   7674162014901917726537932116865512909291406576551328397046176856/
     2903005726942520074922753127299384376977622417179875137701458116859 \
6049368987123841, 
   94690165605482531317925879428708453600918510467044846981979819741/
     8578652648182939165449633595837219514004029975129682670191812458100 \
04058612665408541, 
   74228199976863811380755765181103511603220418505530775649561987369792 2/
     16102368177931016000604389365298154821226053794522504987750812030546 \
303860462622639468467, 
   15644209801006165966614379673923067197820002213040052573521720666049 4/
     81244506782483083764424573238460862040361990328278361869440396392746 \
26518069614456111997, 
   54508615141031158027287162454904665119075334148200203336974401449250 459/
     677550076156312589151606847313466421029568706768017073431505419128 \
5227373100355636335460117, 
   57298151398852165521498313842152765925302311161031467997423588454466 11/
     1704402855350628845479941739022307947335817564963935165047026822493 \
670450726897758102240653, 
   15448341828485371723604907763733894798396999714411747749815784481035 73827/
     1099490388325602979491260965419207631149978994675615122100825413 \
287691329591468464503474756637, 
   14385499644470087221892059537063202666733970301905272321850948727986 \
7076929/24492765297582696960784736917897289995088061487273849808210740 \
2360028650891934769122038759022607, 
   21147305933613335975703724335208049162725414743261318438707033670322 91239/
     8611906291558613638586649800147726185841209509249081342862553775 \
852392623120673568795731901097, 
   14615656974740672346218082529732663691850335975882207460482126160194 74979/
     1423392226824376200799039744416107859202834924822474275363336039 \
5103212733169777975724696346917, 
   22818336988249035656534678070253095937092355453761495618797584340543 \
88759353027/
     5313556211244822874230576764031644946866140195246164289385437274534 \
4155017675385551493728176913537421, 
   45475190386894703927321163053745931297738574246576218449276061029877 \
90196453/2531664726627963632966188114948646304667731511735495235440408 \
00591757243381243218278753467363146339, 
   84054369879675844974934677741726105090465425461757594304975213092696 \
71908251/1118561448341265560270437703543079199403408482266929412518364 \
367217988536837236035739912572518720573, 
   21955654046493313002896634958013006674522479333545761029769242176754 \
63818765245177/
     6983182535790998199080637074428672543906480474479760371415725881623 \
65588828704657277528396814639198437631, 
   44067457226898522027770363554464370585269408680749430046819954134674 \
929499784597093/
     3349454010286892133471837995127191356686260920770224027739264923548 \
2619782429667798518616809613334632418579, 
   21841232679277943202253146599100069364571950358205265514000908200309 \
5103043544620061603/
     3966665740270629934397159248023399730237078388240978321287163685790 \
61112288738688196578104436637440362534482309, 
   11409689564245096955310683194320211571600968805189164734264780598727 \
28020777433652321/
     4950623795181764306378355978019101062489065090766076039946122490998 \
318028074236870721119332804275135560701015, 
   36771078890632459718859906629883387779485082829466124036783284420048 \
77441571129859942/
     3811334026755339486628517085323008196567952304603877757541811168918 \
4440045899986519603896774691085286460268963, 
   15056452643437982632345115687321173329936655145929814230187482071440 \
949676018726905704748031/
     3727574861776103173507020486272410642538946581966828247371570741199 \
42155221611443648253609018634666134859559598131515, 
   45169892794341895770842483694396664502862499203348008500974578764958 \
32638007432252809348731617471/
     2670766516590101068963588047280235767435762349318095659384398424866 \
77485933660721100386303184370483583607516659459390027765, 
   10210118960170926190310107165158089077772959997803727125748756572211 \
0524363193945756326384093576/
     1441625237416103143169952199923608880982540854631519738916363400438 \
9075139689996288179487526080974538219020487370062803365, 
   43304283224172493326307799406512391195721629930415141316608966694148 \
29642158974354016914226534561/
     1459957173364663992429955081754600137724263717999705834551705732099 \
078640635851477843215444792295999128225902084026535792835, 
   14508936359063149080780883537181149078655785995285797377380426722104 \
434728087984279663716452484482943/
     1167849437018598754608973882541235252202338024578161301244374873527 \
7228164135796822415916251119635147060399985868857214356539945, 
   48304748422100741249066840498803221406951302310119608885288328066800 \
73258461411018544275580710162/
     9281939867573373170833062451163622376474231205724421024748516560104 \
464246463067900060156000923887593568376680246564784680111, 
   23520779384511743636145308774132144300972249140737148050324180131869 \
583324431800097056898701352136/
     1078835305919593701331252668504099715560693430304690902712573482149 \
84674602333035100699190240246496784261951906472367743249159, 
   87323410789451382253920173940307489286585212883680237113479746396109 \
0922272781353100772178001769/
     9559753864203405281725685140123295090223170379293374052586137654154 \
105141910597308264589621873811466546673768385105508807121, 
   79035502431495289706996362052889812583373709654824426894793032502308 \
244673162271250920957647380/
     2064952280660357226843985271923235276990862097379305616390053107230 \
870883877093466309306721333740993219599434371389037804133, 
   90445793185153797800852899676114761492502755265500783536135548655294 \
5476658021761049529147712809897/
     5639079985374152293770378927726325333457666729510004720504670634482 \
3255593534225439032499478751226896469319518453287523933405033, 
   23553060716641396843799712649572408615393589819050441982985896275120 \
20426151676970407951054161695929167/
     3503972336613451404073290700466364645648155714253314102459423921321 \
48806886447027343007366165332219929193819666595827008264726888405};

Length[seq]

(*  75  *)

The sequence spans an incredible range

mm = MinMax[seq // N]

(*  {6.92887*10^-16, 6.70123*10^99}  *)

ListLogPlot[seq,
 Frame -> True,
 Axes -> False,
 PlotRange -> All]

There is a major break at about n = 39

Using the first 30 terms in FindSequenceFunction

AbsoluteTiming[fsf30 = FindSequenceFunction[seq[[;; 30]], n];]

(*  {294.22, Null}  *)

It took almost five minutes and the resulting "simple function" has an extremely large LeafCount

LeafCount@fsf30

(*  10011  *)

fsf30s[n_] = fsf30 // Simplify;

Simplify helps a lot but still leaves a very complicated function

LeafCount@fsf30s[n]

(*  1566  *)

Count[fsf30s[n], #, Infinity] & /@ {_Gamma, _HypergeometricPFQ}

(*  {89, 14}  *)

Checking if the function tracks the sequence

$MaxExtraPrecision = 500;

With[{max = 45},
 (Equal @@@
   Transpose[{
     (Table[fsf30s[n] // N[#, 100] &, {n, max}]),
     (seq[[;; max]] // N)}])]

(*  {True, True, True, True, True, True, True, True, True, True, True, True, \
True, True, True, True, True, True, True, True, True, True, True, True, True, \
True, True, True, True, True, True, True, True, True, True, True, True, True, \
False, False, False, False, False, False, False}  *)

Even with very high precision {100) the function does not align with the sequence for n = 39 and higher, i.e., the function based on the first thirty terms cannot see the break in the sequence.

Using all 75 terms of the sequence

AbsoluteTiming[
 TimeConstrained[
  fsf30 = FindSequenceFunction[seq, n];,
  Infinity]]

(*  {201.327, Null}  *)

fsf30 // Head

(*  FindSequenceFunction  *)

With 75 terms FindSequenceFunction returned unevaluated. It seems unlikely that "a simple function" can be found to handle the break in the sequence. You might want to consider building a Piecewise function by breaking the sequence into segments.