In[979]:= 1 + 1 Out[979]= 2 In[980]:= x + y Out[980]= x + y In[981]:= y + x Out[981]= x + y In[982]:= x^2 + y^2 2 2 Out[982]= x + y In[983]:= {1,2,3} Out[983]= {1, 2, 3} In[984]:= {1,3,2} Out[984]= {1, 3, 2} In[985]:= {1,3,2,2,2} Out[985]= {1, 3, 2, 2, 2} In[986]:= Union[{1,34,5,4,3},{435,2,3,3}] Out[986]= {1, 2, 3, 4, 5, 34, 435} In[987]:= f /@ {1,2,3,4,5} Out[987]= {f[1], f[2], f[3], f[4], f[5]} In[988]:= f@@{1,2,3,4,5} Out[988]= f[1, 2, 3, 4, 5] In[989]:= f[{a,b,c}] Out[989]= f[{a, b, c}] In[990]:= Pi // Sin Out[990]= 0 In[991]:= Sin[Pi] Out[991]= 0 In[992]:= {1,2,3} + {4,5,6} Out[992]= {5, 7, 9} In[993]:= 5 {1,2,3} Out[993]= {5, 10, 15} In[994]:= {1,2,3}^2 Out[994]= {1, 4, 9} In[995]:= {1,2,3}{1,2,3} Out[995]= {1, 4, 9} In[996]:= {1,2,3}.{1,2,3} Out[996]= 14 In[997]:= 5 3 Out[997]= 15 In[998]:= 5x Out[998]= 5 x In[999]:= 53 Out[999]= 53 In[1000]:= 5 3 Out[1000]= 15 In[1001]:= {{1,2},{3,4}} Out[1001]= {{1, 2}, {3, 4}} In[1002]:= {{1,2},{3,4}} // MatrixForm Out[1002]//MatrixForm= 1 2 3 4 In[1003]:= {{1,2},{3,4}} . {{1,2},{3,4}} Out[1003]= {{7, 10}, {15, 22}} In[1004]:= {{1,2},{3,4}} * {{1,2},{3,4}} Out[1004]= {{1, 4}, {9, 16}} In[1005]:= Inverse[{{1,2},{3,4}} ] 3 1 Out[1005]= {{-2, 1}, {-, -(-)}} 2 2 In[1006]:= Nullspace[{{1,2},{3,4}} ] Out[1006]= Nullspace[{{1, 2}, {3, 4}}] In[1007]:= NullSpace[{{1,2},{3,4}} ] Out[1007]= {} In[1008]:= NullSpace[{{1,2,3},{3,4,5},{7,8,9}} ] Out[1008]= {{1, -2, 1}} In[1009]:= Plot[x^2, {x, -1, 1}] Out[1009]= -Graphics- In[1010]:= /tmp/m00002629431 is a 360x226 JPEG image, color space YCbCr, 3 comps, Huffman coding. Building XImage...done In[1010]:= {1 + x, 2 x, x^2} /. x -> 1 Out[1010]= {2, 2, 1} In[1011]:= {1 + x, 2 x, x^2} /. x -> x^2 2 2 4 Out[1011]= {1 + x , 2 x , x } In[1012]:= {1 + x, 2 x, x^2} /. n_Integer -> n^2 4 Out[1012]= {1 + x, 4 x, x } In[1013]:= {1 + x, 2 x, x^2} /. n_Integer -> n^5 32 Out[1013]= {1 + x, 32 x, x } In[1014]:= {1 + x, 2 x, x^2} /. n_Integer -> 2 n 4 Out[1014]= {2 + x, 4 x, x } In[1015]:= f[x_] := x^2 In[1016]:= f[x] 2 Out[1016]= x In[1017]:= f[y] 2 Out[1017]= y In[1018]:= Clear[f] In[1019]:= f[5] Out[1019]= f[5] In[1020]:= f[x] Out[1020]= f[x] In[1021]:= f[x] := x^2 In[1022]:= f[x] 2 Out[1022]= x In[1023]:= f[y] Out[1023]= f[y] In[1024]:= f[5] Out[1024]= f[5] In[1025]:= Clear[f] In[1026]:= SetOptions[$Output, PageWidth-> Infinity]; Block[{Short=Identity},Get["/home/mkauers/test.m"]]; SetOptions[$Output, PageWidth-> 142]; In[1027]:= f[5] Out[1027]= 20 In[1028]:= Table[x^2, {x, -5, 5}] Out[1028]= {25, 16, 9, 4, 1, 0, 1, 4, 9, 16, 25} In[1029]:= Do[ If[ PrimeQ[n], Print[n] ] , {n, 1, 20}] 2 3 5 7 11 13 17 19 In[1030]:= x == 0 Out[1030]= x == 0 In[1031]:= Which Out[1031]= Which In[1032]:= CylindricalDecomposition[ {x > 0, y > 0}, {x,y} ] Out[1032]= x > 0 && y > 0 In[1033]:= CylindricalDecomposition[ {x^2 + 1 > 0, y > 0}, {x,y} ] Out[1033]= y > 0 In[1034]:= CylindricalDecomposition[ {x>0,y>0,z>0,x y z == 1, (x^10+y^10+z^10)^2 >= 3 (x^13 +y^13+z^13)}, {x,y,z} ] C-c C-c Interrupt> a Out[1034]= $Aborted In[1035]:= CylindricalDecomposition[ {x>0,y>0,z>0,x y z == 1, (x^10+y^10+z^10)^2 < 3 (x^13 +y^13+z^13)}, {x,y,z} ] C-c C-c Interrupt> a Out[1035]= $Aborted In[1036]:= {x>0,y>0,z>0,x y z == 1, (x^10+y^10+z^10)^2 < 3 (x^13 +y^13+z^13)} /. z -> 1/(x y) 1 10 1 10 2 13 1 13 Out[1036]= {x > 0, y > 0, --- > 0, True, (x + ------- + y ) < 3 (x + ------- + y )} x y 10 10 13 13 x y x y In[1037]:= CylindricalDecomposition[%, {x,y}] Out[1037]= False In[1038]:= Table[ff1[n], {n, 0, 10}] Out[1038]= {2, 4, 17, 89, 509, 3045, 18696, 116709, 736901, 4691687, 30061811} In[1039]:= Table[ff1[n], {n, 0, 100}] Out[1039]= {2, 4, 17, 89, 509, 3045, 18696, 116709, 736901, 4691687, 30061811, 193595506, 1251885712, 8123168344, 52862903520, 344877120429, > 2254884271317, 14771198731515, 96926826217305, 636985736584890, 4191851069925915, 27619459868630055, 182183259464324040, > 1202937318892802610, 7950262951993175424, 52588552003095295080, 348131441297221087436, 2306279517035401918724, 15288835899817040637872, > 101416868969270383409728, 673132978321566673322368, 4470229466608958508982141, 29701747828550731286954613, 197443926317438529637816803, > 1313114785928089497929994789, 8736699119981775011875120042, 58152371715884905252907989057, 387215996008591695477472511389, > 2579266129073146030381995823916, 17186518561858316638321961262110, 114556848247587292785386299001291, 763815805697636852963718733430613, > 5094296466543607890655128603639615, 33986135441805374799796816653889500, 226796193798144623822444557851770040, > 1513843358002308769926069953305474920, 10107237020884383800309252266994947920, 67497197126403128196300641136643253490, > 450854633173833361396323517234754988000, 3012179439603057011477574568029889452072, 20128660909733910555897990541582939522596, > 134534952009374985314908097167605281792076, 899369670779603468018579763288372458547612, 6013413491471302867605588605176042460563740, > 40214312445560620610484191378660585594206224, 268976397071239731484411708238926714733482692, > 1799363822299441954223061953493860660738214672, 12039059761216321641274475997176783654005548864, > 80562427793422609090906338952776097596023977320, 539183567152849852723643179462804847160027263112, > 3609131684164726179073923467797767201573492504096, 24161769937627907065985439512400888456170083695936, > 161775501313307212591057304566799111405360515149312, 1083310385556310097571010654931662240479159730723869, > 7255167425905233516643950859386087912107022583841205, 48595345037700133736229973792566634774467454584394419, > 325531482060318796091715207317993624551021458775673277, 2180925763550783524491845017314339156385060688315027850, > 14612915336628011246643379641058788396813907018770837645, 97921836665471423902183179718375253211596819188888307625, > 656247392115319691743498792427440562094777509364080106180, 4398450456113647052217873525036906383522550334660436306790, > 29483203013663884145337553072398902473215581110797429613425, 197647482536284889416277808051989462463156807366529121071215, > 1325099365818280656620916648658271764921201412930257619040765, 8884746781389900047391615301193030906309517788416643301716604, > 59577197548649091870678951321207358165997059834340075522913372, 399532848852476148296912512185144024834111162638717576595043500, > 2679546858259237056069033163641273188712182417648215688725110696, 17972385966184465910814968267684507127117335655379402226235740310, > 120554865392512357303319440413444826412725231110429391938885564587, 808718755397067598644664917205254552365527910744227640113786670801, > 5425543228823511265172918592872322074826236630065088895681199221725, > 36401651262301534069987659789793277427322062777522252874107328228984, > 244247606577140853602431061688469485082372041097201263331392070412815, > 1638966901551711681259194748409403210970184886607134221372046324884995, > 10998664870143840772478168321661560789925667901851281708084434604807120, > 73814041494467434484608658574605066783979873300192602761981042911722940, > 495412046417315555902205386761014913166127474214148986293586466180529880, > 3325232827266860934635742865773734471301794316169684154873482051051802200, > 22320547950030031517667820167313985967413768411341061852955359169775461280, > 149835367964235354553181335432264638999371664947788366688512222218550251640, > 1005888718469808799499473532532834964613000207313879825349097522740881191600, > 6753222899257724186072882776634775137674078283877134196321401069241014406800, > 45341641061007885336233851934168561703835737322388613938361959741085701021600, > 304444308718072327505745962426415325997600203815245467252282381717631507963250, > 2044289737254320417729598168640990499317676055687288741865266946300834485962400, > 13727786139571972187109462119575084388064089555670774101699083425127107645666600, > 92189518913439533158582152486573291491723179095782845127304671572270168128134100, > 619134655554648518336209157902428177360153111905199597520420548489024980190593340, > 4158251463258564744783384422846352670411962692913718782763666078142277867069917940} In[1040]:= ff1[1000] Out[1040]= 3109469514424898096038263456104294354714830712284680835308026792729512845881687637816876000971488329418780089738343684813804382056\ > 2503080597606788407060976081436729679649548000140524057741704981408582049244819077473866952403446479538080910494909457531418919832237132\ > 1017166605043744565105225763247300243389336979268926307868413297161377770028369660082489935631311644083157010454978792408351560378793761\ > 8886425470560965228510177811577376956761983027145870690781342232247468596271152744400640712523324005604824091043450927782544084630375944\ > 2491296168142575757247398503676358577097680111272411818756965234910980200665751752995244638920828614717712713181806347341516461389979387\ > 2917830168764084582588619258410885343085007004739398711809159355765984429977707808544286115778475349109927909499535159526307470858388535\ > 772501400671437216 In[1041]:= (* hypothesis: ff1[n] ~ phi^n *(1 + ...) ; question: what is phi? *) In[1042]:= ff1[n+1]/ff1[n] /. n -> 100 // N Out[1042]= 6.71657 In[1043]:= Table[N[ff1[n+1]/ff1[n]], {n, 1000, 1100, 10}] Out[1043]= {6.74663, 6.74666, 6.74669, 6.74673, 6.74676, 6.74679, 6.74682, 6.74685, 6.74688, 6.74691, 6.74693} In[1044]:= Do[Print[N[ff1[n+1]/ff1[n]]], {n, 1000, 1100, 10}] 6.74663 6.74666 6.74669 6.74673 6.74676 6.74679 6.74682 6.74685 6.74688 6.74691 6.74693 In[1045]:= Do[Print[N[ff1[n+1]/ff1[n]]], {n, 10000, 10100, 10}] 6.74966 6.74966 6.74966 6.74966 6.74966 6.74966 6.74966 6.74966 6.74967 6.74967 6.74967 In[1046]:= Do[Print[N[ff1[n+1]/ff1[n]]], {n, 20000, 20100, 10}] 6.74983 6.74983 6.74983 6.74983 6.74983 6.74983 6.74983 6.74983 6.74983 6.74983 6.74983 In[1047]:= (* ff1[n+1]/ff1[n] ~ phi( 1 + ???/n + ???/n^2 + ???/n^3 + ... ) *) In[1048]:= u[n_Integer] := ff1[n+1]/ff1[n] In[1049]:= u[1000] // N Out[1049]= 6.74663 In[1050]:= N[2u[2n]-u[n] /. n -> 10000] Out[1050]= 6.75 In[1051]:= N[2u[2n]-u[n] /. n -> 10000, 20] Out[1051]= 6.7499999840648200459 In[1052]:= Rationalize[6.7499999] Out[1052]= 6.75 In[1053]:= Rationalize[6.749999] Out[1053]= 6.75 In[1054]:= 675/100 27 Out[1054]= -- 4 In[1055]:= ff1[n]/(27/4)^n /. n -> 100 // N Out[1055]= 0.0488128 In[1056]:= ff1[n]/(27/4)^n /. n -> 100 // N Out[1056]= 0.0488128 In[1057]:= ff1[n]/(27/4)^n /. n -> 1000 // N Out[1057]= 0.0154495 In[1058]:= ff1[n]/(27/4)^n /. n -> 10000 // N Out[1058]= 0.00488598 In[1059]:= ff1[n]/(27/4)^n /. n -> 100000 // N Out[1059]= 0.0015451 In[1060]:= ff1[n]/(27/4)^n /. n -> 1000000 // N Out[1060]= 0.000488602 In[1061]:= (n+1)^alpha/n^alpha // FullSimplify alpha (1 + n) Out[1061]= ------------ alpha n In[1062]:= phi^n n^alpha alpha n Out[1062]= n phi In[1063]:= (% /. n -> n +1)/% alpha (1 + n) phi Out[1063]= ---------------- alpha n In[1064]:= Log[ff1[n]/(27/4)^n]/Log[n] /. n -> 100 // N Out[1064]= -0.655733 In[1065]:= Log[ff1[n]/(27/4)^n]/Log[n] /. n -> 200 // N Out[1065]= -0.635268 In[1066]:= Log[ff1[n]/(27/4)^n]/Log[n] /. n -> 1000 // N Out[1066]= -0.603696 In[1067]:= Log[ff1[n]/(27/4)^n]/Log[n] /. n -> 10000 // N Out[1067]= -0.577762 In[1068]:= Log[ff1[n]/(27/4)^n]/Log[n] /. n -> 100000 // N Out[1068]= -0.562209 In[1069]:= u[n_Integer] := Log[ff1[n]/(27/4)^n]/Log[n] In[1070]:= u[100] // N Out[1070]= -0.655733 In[1071]:= 2u[2n]-u[n] /. n -> 10000 // N Out[1071]= -0.566876 In[1072]:= 2u[2n]-u[n] /. n -> 100000 // N Out[1072]= -0.555144 In[1073]:= 2u[2n]-u[n] /. n -> 1000000 // N Out[1073]= -0.546887 In[1074]:= D[n^alpha, n] -1 + alpha Out[1074]= alpha n In[1075]:= D[n^alpha, n]/(n^alpha) alpha Out[1075]= ----- n In[1076]:= n D[n^alpha, n]/(n^alpha) Out[1076]= alpha In[1077]:= n^n n Out[1077]= n In[1078]:= ff1[n]/(27/4)^n ff1[n] Out[1078]= ------ 27 n (--) 4 In[1079]:= (* ff1[n]/(27/4)^n ~ n^alpha *) In[1080]:= gu[n_] = ff1[n]/(27/4)^n ff1[n] Out[1080]= ------ 27 n (--) 4 In[1081]:= n (gu[n+1] - gu[n])/ gu[n] /. n -> 100 // N Out[1081]= -0.495323 In[1082]:= n (gu[n+1] - gu[n])/ gu[n] /. n -> 1000 // N Out[1082]= -0.499528 In[1083]:= n (gu[n+1] - gu[n])/ gu[n] /. n -> 10000 // N Out[1083]= -0.499953 In[1084]:= n (gu[n+1] - gu[n])/ gu[n] /. n -> 100000 // N Out[1084]= -0.499995 In[1085]:= u[n_] := n (gu[n+1] - gu[n])/ gu[n] In[1086]:= 2u[2n] - u[n] /. n -> 10000 // N Out[1086]= -0.5 In[1087]:= N[2u[2n] - u[n] /. n -> 10000, 100] Out[1087]= -0.4999999977086718358114741780223039381767401567671506177430644979671568406113580957121239543982650205 In[1088]:= ff1[n]/ ((27/4)^n n^(-1/2)) /. n -> 100 // N Out[1088]= 0.488128 In[1089]:= ff1[n]/ ((27/4)^n n^(-1/2)) /. n -> 1000 // N Out[1089]= 0.488555 In[1090]:= ff1[n]/ ((27/4)^n n^(-1/2)) /. n -> 10000 // N Out[1090]= 0.488598 In[1091]:= ff1[n]/ ((27/4)^n n^(-1/2)) /. n -> 100000 // N Out[1091]= 0.488602 In[1092]:= ff1[n]/ ((27/4)^n n^(-1/2)) /. n -> 200000 // N Out[1092]= 0.488602 In[1093]:= ff1[n]/ ((27/4)^n n^(-1/2)) /. n -> 300000 // N Out[1093]= 0.488602 In[1094]:= %^2 Out[1094]= 0.238732 In[1095]:= % Pi Out[1095]= 0.75 In[1096]:= Sqrt[3/4 / Pi] 3 Sqrt[--] Pi Out[1096]= -------- 2 In[1097]:= N[%, 20] Out[1097]= 0.48860251190291992159 In[1098]:= %1096 (27/4)^n n^(-1/2) -1 - 2 n 1/2 + 3 n 2 3 Out[1098]= -------------------- Sqrt[n] Sqrt[Pi] In[1099]:= Table[ff2[n], {n,1,10}] Out[1099]= {5, 8, 11, 14, 17, 20, 23, 26, 29, 32} In[1100]:= InterpolatingPolynomial[%, n] Out[1100]= 5 + 3 (-1 + n) In[1101]:= Expand[%] Out[1101]= 2 + 3 n In[1102]:= Table[%, {n,1,10}] Out[1102]= {5, 8, 11, 14, 17, 20, 23, 26, 29, 32} In[1103]:= Table[%%, {n,1,20}] Out[1103]= {5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35, 38, 41, 44, 47, 50, 53, 56, 59, 62} In[1104]:= Table[ff1[n],{n,1,10}] Out[1104]= {4, 17, 89, 509, 3045, 18696, 116709, 736901, 4691687, 30061811} In[1105]:= InterpolatingPolynomial[Table[ff1[n],{n,1,10}], n] 59 289 493 323 917 793 1204361 1314647 (-9 + n) Out[1105]= 4 + (13 + (-- + (--- + (--- + (--- + (--- + (--- + (------- + ----------------) (-8 + n)) (-7 + n)) (-6 + n)) (-5 + n)) (-4 + n)) 2 6 8 5 16 18 40320 72576 > (-3 + n)) (-2 + n)) (-1 + n) In[1106]:= Expand[%] 2 3 4 5 6 7 2846139161 n 91788995107 n 1030581521801 n 12341509489 n 17689330159 n 224705063 n 127249723 n Out[1106]= -5556161 + ------------ - -------------- + ---------------- - -------------- + -------------- - ------------ + ------------ - 180 5040 90720 2880 17280 1440 8640 8 9 5276969 n 1314647 n > ---------- + ---------- 6720 72576 In[1107]:= % /. n -> 20 Out[1107]= 710654232673 In[1108]:= ff1[20] Out[1108]= 4191851069925915 In[1109]:= << Guess.m q$::shdw: Symbol q$ appears in multiple contexts {RISCComb`, Global`}; definitions in context RISCComb` may shadow or be shadowed by other definitions. SetDelayed::write: Tag Symmetrize in Symmetrize[term_, (f_)[nn__]] is Protected. Guess Package by Manuel Kauers — © RISC Linz — V 0.52 2015-04-30 In[1110]:= GuessMinRE[Table[ff2[n], {n,0,100}], f[n]] Out[1110]= GuessMinRE[{2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35, 38, 41, 44, 47, 50, 53, 56, 59, 62, 65, 68, 71, 74, 77, 80, 83, 86, 89, > 92, 95, 98, 101, 104, 107, 110, 113, 116, 119, 122, 125, 128, 131, 134, 137, 140, 143, 146, 149, 152, 155, 158, 161, 164, 167, 170, > 173, 176, 179, 182, 185, 188, 191, 194, 197, 200, 203, 206, 209, 212, 215, 218, 221, 224, 227, 230, 233, 236, 239, 242, 245, 248, 251, 2 > 254, 257, 260, 263, 266, 269, 272, 275, 278, 281, 284, 287, 290, 293, 296, 299, 302}, -5 + n ] In[1111]:= f[n] 2 Out[1111]= -5 + n In[1112]:= Clear[f] In[1113]:= GuessMinRE[Table[ff2[n], {n,0,100}], f[n]] 5 2 Out[1113]= (-(-) - n) f[n] + (- + n) f[1 + n] 3 3 In[1114]:= GuessMinRE[Table[ff1[n], {n,0,100}], f[n]] 2 3 4 5 324 2619 n 15909 n 22947 n 16305 n 5319 n 6 Out[1114]= (--- + ------ + -------- + -------- + -------- + ------- + 27 n ) f[n] + 11 11 22 22 22 22 2 3 4 5 6 468 3117 n 7360 n 32407 n 17931 n 4769 n 43 n > (-(---) - ------ - ------- - -------- - -------- - ------- - -----) f[1 + n] + 11 11 11 44 44 44 4 2 3 4 5 72 468 n 1033 n 187 n 507 n 241 n 6 > (-- + ----- + ------- + ------ + ------ + ------ + n ) f[2 + n] 11 11 11 2 11 22 In[1115]:= << Asymptotics.m Asymptotics Package by Manuel Kauers — © RISC Linz — V 0.11 (2012-07-19) In[1116]:= Asymptotics[%1114, f[n]] 27 n n (--) 4 4 Out[1116]= {----, -------} 3/2 Sqrt[n] n In[1117]:= Asymptotics[%1114, f[n], Order -> 5] n 295911 36939 1155 145 9 27 n 37195739 187103 6425 49 7 4 (1 - --------- + -------- - ------- + ------ - ---) (--) (1 - -------------- - ------------ + ---------- + -------- - ----) 5 4 3 2 8 n 4 5 4 3 2 72 n 262144 n 32768 n 1024 n 128 n 46438023168 n 644972544 n 2239488 n 10368 n Out[1117]= {------------------------------------------------------, ------------------------------------------------------------------------} 3/2 Sqrt[n] n In[1118]:= Last[%] 27 n 37195739 187103 6425 49 7 (--) (1 - -------------- - ------------ + ---------- + -------- - ----) 4 5 4 3 2 72 n 46438023168 n 644972544 n 2239488 n 10368 n Out[1118]= ------------------------------------------------------------------------ Sqrt[n] In[1119]:= ff1[n] / % 4 n (--) Sqrt[n] ff1[n] 27 Out[1119]= ---------------------------------------------------------------- 37195739 187103 6425 49 7 1 - -------------- - ------------ + ---------- + -------- - ---- 5 4 3 2 72 n 46438023168 n 644972544 n 2239488 n 10368 n In[1120]:= N[%1119 /. n -> 1000 , 20] Out[1120]= 0.48860251190291992159 In[1121]:= Sqrt[3/4 / Pi] 3 Sqrt[--] Pi Out[1121]= -------- 2 In[1122]:= N[%, 20] Out[1122]= 0.48860251190291992159 In[1123]:= N[%1119 /. n -> 1000 , 20] Out[1123]= 0.48860251190291992159 In[1124]:= N[%1119 /. n -> 1000 , 100] Out[1124]= 0.4886025119029199215864256471590581598962415546851037985697054277948905372267404280289315998381084633 In[1125]:= N[Sqrt[3/4 / Pi], 100] Out[1125]= 0.4886025119029199215863846228383470045758856081942277021382431574458410003616365304056148187039700424 In[1126]:= % - %% -23 Out[1126]= -4.10243207111553203559464908760964314622703490495368651038976233167811341384209 10 In[1127]:= Asymptotics[%1114, f[n], Order -> 20] n 500107926698037066995963 12322967114104483154175 1744050969382356538475 105666176941167049515 Out[1127]= {(4 (1 - ---------------------------- + -------------------------- + -------------------------- - ------------------------ + 20 19 18 17 302231454903657293676544 n 9444732965739290427392 n 1180591620717411303424 n 73786976294838206464 n 9898775194885242435 77871582933450075 10265348737564125 640669821436755 79183820728095 > ----------------------- - --------------------- + -------------------- - ------------------- + ------------------ - 16 15 14 13 12 9223372036854775808 n 72057594037927936 n 9007199254740992 n 562949953421312 n 70368744177664 n 2475570313125 310496335695 19402289907 2421696563 37844235 4735445 295911 36939 1155 > ----------------- + ---------------- - -------------- + ------------- - ----------- + ---------- - --------- + -------- - ------- + 11 10 9 8 7 6 5 4 3 2199023255552 n 274877906944 n 17179869184 n 2147483648 n 33554432 n 4194304 n 262144 n 32768 n 1024 n 145 9 3/2 27 n 3276752143173355454938876142411193697204274843 > ------ - ---)) / n , ((--) (1 - --------------------------------------------------- + 2 8 n 4 20 128 n 24107928143612940505683647991134236454770704384 n 116486209978833050055373789455863858714203825 20411949498621999142098821061996793391915 > ------------------------------------------------ + ----------------------------------------------- - 19 18 83708083831989376755845999969216098801287168 n 1162612275444296899386749999572445816684544 n 161028983039249941722643650692199641405 36051276856715405802335064240402485 > ------------------------------------------- - ------------------------------------------ + 17 16 897077373028006866810763888558985969664 n 12459407958722317594593942896652582912 n 319278910130903213009400265181075 31527643079491604711912403815 2225984523401385764053525895 > --------------------------------------- + ------------------------------------ - ---------------------------------- - 15 14 13 10815458297502011800862797653344256 n 50071566192138943522512952098816 n 347719209667631552239673278464 n 914762790861043654466515 10699973519234898176005 6523794786566402527 453012721858735181 > -------------------------------- + ----------------------------- + --------------------------- - ------------------------ - 12 11 10 9 4829433467605993781106573312 n 5589622068988418728132608 n 77633639847061371224064 n 539122498937926189056 n 50429081275573 859900796755 1672286429 37195739 187103 6425 49 7 > --------------------- + ------------------- + ----------------- - -------------- - ------------ + ---------- + -------- - ----)) / 8 7 6 5 4 3 2 72 n 831979165027663872 n 1444408272617472 n 20061226008576 n 46438023168 n 644972544 n 2239488 n 10368 n > Sqrt[n]} In[1128]:= Last[%] 27 n 3276752143173355454938876142411193697204274843 116486209978833050055373789455863858714203825 Out[1128]= ((--) (1 - --------------------------------------------------- + ------------------------------------------------ + 4 20 19 24107928143612940505683647991134236454770704384 n 83708083831989376755845999969216098801287168 n 20411949498621999142098821061996793391915 161028983039249941722643650692199641405 > ----------------------------------------------- - ------------------------------------------- - 18 17 1162612275444296899386749999572445816684544 n 897077373028006866810763888558985969664 n 36051276856715405802335064240402485 319278910130903213009400265181075 31527643079491604711912403815 > ------------------------------------------ + --------------------------------------- + ------------------------------------ - 16 15 14 12459407958722317594593942896652582912 n 10815458297502011800862797653344256 n 50071566192138943522512952098816 n 2225984523401385764053525895 914762790861043654466515 10699973519234898176005 > ---------------------------------- - -------------------------------- + ----------------------------- + 13 12 11 347719209667631552239673278464 n 4829433467605993781106573312 n 5589622068988418728132608 n 6523794786566402527 453012721858735181 50429081275573 859900796755 1672286429 > --------------------------- - ------------------------ - --------------------- + ------------------- + ----------------- - 10 9 8 7 6 77633639847061371224064 n 539122498937926189056 n 831979165027663872 n 1444408272617472 n 20061226008576 n 37195739 187103 6425 49 7 > -------------- - ------------ + ---------- + -------- - ----)) / Sqrt[n] 5 4 3 2 72 n 46438023168 n 644972544 n 2239488 n 10368 n In[1129]:= ff1[n] / % 4 n Out[1129]= ((--) Sqrt[n] ff1[n]) / 27 3276752143173355454938876142411193697204274843 116486209978833050055373789455863858714203825 > (1 - --------------------------------------------------- + ------------------------------------------------ + 20 19 24107928143612940505683647991134236454770704384 n 83708083831989376755845999969216098801287168 n 20411949498621999142098821061996793391915 161028983039249941722643650692199641405 > ----------------------------------------------- - ------------------------------------------- - 18 17 1162612275444296899386749999572445816684544 n 897077373028006866810763888558985969664 n 36051276856715405802335064240402485 319278910130903213009400265181075 31527643079491604711912403815 > ------------------------------------------ + --------------------------------------- + ------------------------------------ - 16 15 14 12459407958722317594593942896652582912 n 10815458297502011800862797653344256 n 50071566192138943522512952098816 n 2225984523401385764053525895 914762790861043654466515 10699973519234898176005 6523794786566402527 > ---------------------------------- - -------------------------------- + ----------------------------- + --------------------------- - 13 12 11 10 347719209667631552239673278464 n 4829433467605993781106573312 n 5589622068988418728132608 n 77633639847061371224064 n 453012721858735181 50429081275573 859900796755 1672286429 37195739 187103 > ------------------------ - --------------------- + ------------------- + ----------------- - -------------- - ------------ + 9 8 7 6 5 4 539122498937926189056 n 831979165027663872 n 1444408272617472 n 20061226008576 n 46438023168 n 644972544 n 6425 49 7 > ---------- + -------- - ----) 3 2 72 n 2239488 n 10368 n In[1130]:= N[%1129 /. n -> 1000, 100] Out[1130]= 0.4886025119029199215863846228383470045758856081942277021382431509004861360852787222927448020114888204 In[1131]:= N[Sqrt[3/4 / Pi], 100] Out[1131]= 0.4886025119029199215863846228383470045758856081942277021382431574458410003616365304056148187039700424 In[1132]:= % - %% -63 Out[1132]= 6.5453548642763578081128700166924812221 10 In[1133]:= GuessMinRE[Table[Prime[n],{n,1,100}],f[n]] Throw::nocatch: Uncaught Throw[No recurrence found.] returned to top level. Out[1133]= Hold[Throw[No recurrence found.]] In[1134]:= Head[%] Out[1134]= Hold In[1135]:= GuessMinRE[Table[ff1[n],{n,1,100}],f[n]] 2 3 5 33480 100638 n 122340 n 153237 n 4 8883 n 6 Out[1135]= (----- + -------- + --------- + --------- + 2355 n + ------- + 27 n ) f[n] + 11 11 11 22 22 2 3 4 5 6 24840 66744 n 72258 n 161281 n 48871 n 7607 n 43 n > (-(-----) - ------- - -------- - --------- - -------- - ------- - -----) f[1 + n] + 11 11 11 44 44 44 4 2 3 4 5 3240 1551 n 8963 n 2549 n 373 n 6 > (---- + 756 n + ------- + ------- + ------- + ------ + n ) f[2 + n] 11 2 22 22 22 In[1136]:= Cases[%, f[n+i_] -> i, {0, Infinity}] Out[1136]= {1, 2} In[1137]:= Max[%] Out[1137]= 2 In[1138]:= %1135 /. n -> n - 2 2 3 5 33480 100638 (-2 + n) 122340 (-2 + n) 153237 (-2 + n) 4 8883 (-2 + n) 6 Out[1138]= (----- + --------------- + ---------------- + ---------------- + 2355 (-2 + n) + -------------- + 27 (-2 + n) ) f[-2 + n] + 11 11 11 22 22 2 3 4 5 6 24840 66744 (-2 + n) 72258 (-2 + n) 161281 (-2 + n) 48871 (-2 + n) 7607 (-2 + n) 43 (-2 + n) > (-(-----) - -------------- - --------------- - ---------------- - --------------- - -------------- - ------------) f[-1 + n] + 11 11 11 44 44 44 4 2 3 4 5 3240 1551 (-2 + n) 8963 (-2 + n) 2549 (-2 + n) 373 (-2 + n) 6 > (---- + 756 (-2 + n) + -------------- + -------------- + -------------- + ------------- + (-2 + n) ) f[n] 11 2 22 22 22 In[1139]:= Solve[% == 0, f[n]] 2 3 5 33480 100638 (-2 + n) 122340 (-2 + n) 153237 (-2 + n) 4 8883 (-2 + n) 6 Out[1139]= {{f[n] -> (-((----- + --------------- + ---------------- + ---------------- + 2355 (-2 + n) + -------------- + 27 (-2 + n) ) 11 11 11 22 22 2 3 4 5 6 24840 66744 (-2 + n) 72258 (-2 + n) 161281 (-2 + n) 48871 (-2 + n) 7607 (-2 + n) 43 (-2 + n) > f[-2 + n]) - (-(-----) - -------------- - --------------- - ---------------- - --------------- - -------------- - ------------) 11 11 11 44 44 44 4 2 3 4 5 3240 1551 (-2 + n) 8963 (-2 + n) 2549 (-2 + n) 373 (-2 + n) 6 > f[-1 + n]) / (---- + 756 (-2 + n) + -------------- + -------------- + -------------- + ------------- + (-2 + n) )}} 11 2 22 22 22 In[1140]:= f[n] /. % 2 3 5 33480 100638 (-2 + n) 122340 (-2 + n) 153237 (-2 + n) 4 8883 (-2 + n) 6 Out[1140]= {(-((----- + --------------- + ---------------- + ---------------- + 2355 (-2 + n) + -------------- + 27 (-2 + n) ) f[-2 + n]) - 11 11 11 22 22 2 3 4 5 6 24840 66744 (-2 + n) 72258 (-2 + n) 161281 (-2 + n) 48871 (-2 + n) 7607 (-2 + n) 43 (-2 + n) > (-(-----) - -------------- - --------------- - ---------------- - --------------- - -------------- - ------------) f[-1 + n]) / 11 11 11 44 44 44 4 2 3 4 5 3240 1551 (-2 + n) 8963 (-2 + n) 2549 (-2 + n) 373 (-2 + n) 6 > (---- + 756 (-2 + n) + -------------- + -------------- + -------------- + ------------- + (-2 + n) )} 11 2 22 22 22 In[1141]:= % /. n -> 11 -656770464 f[9] 291171888 f[10] 11 (--------------- + ---------------) 11 11 Out[1141]= {--------------------------------------} 29256552 In[1142]:= {f[0],f[1],f[2]} f[3] SetOptions[$Output, PageWidth-> Infinity]; Block[{Short=Identity},Get["/home/mkauers/test.m"]]; SetOptions[$Output, PageWidth-> 142]; Thread::tdlen: Objects of unequal length in {f[0], f[1], f[2]} f[3] {{BinaryFormat -> False, FormatType -> OutputForm, PageWidth -> Infinity, PageHeight -> 22, TotalWidth -> Infinity, TotalHeight -> Infinity, CharacterEncoding :> $CharacterEncoding, NumberMarks :> $NumberMarks, Method -> {File, BinaryFormat -> False}}} cannot be combined. In[1143]:= SetOptions[$Output, PageWidth-> Infinity]; Block[{Short=Identity},Get["/home/mkauers/test.m"]]; SetOptions[$Output, PageWidth-> 142]; In[1144]:= guesscontinue[{1,2,3,4,5}] Throw::nocatch: Uncaught Throw[No recurrence found.] returned to top level. Out[1144]= Hold[Throw[No recurrence found.]] In[1145]:= guesscontinue[{1,2,3,4,5,6,7,8}] Out[1145]= 9 In[1146]:= guesscontinue[{0,1,1,2,3,5,8,13}] Out[1146]= 21 In[1147]:= guesscontinue[{1,2,4,8,16,32}] Out[1147]= 64 In[1148]:=