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m →Equivalent S-expressions: new equivalent S-expressions
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→Sk2 * S(k + 1) and S(k - 1) * Sk2 (lopsided commas): split into two tables to aid readability and clarity of the pattern
Line 1,175: Line 1,175:
S(''k''-1) * S''k''<sup>2</sup> = (''k''/(''k''-2)) / ((''k''+1)/''k'')<sup>2</sup>  through [-1, 0, 3, -2] = [-1, 0, 1, 0] - [0, 0, -2, 2].
S(''k''-1) * S''k''<sup>2</sup> = (''k''/(''k''-2)) / ((''k''+1)/''k'')<sup>2</sup>  through [-1, 0, 3, -2] = [-1, 0, 1, 0] - [0, 0, -2, 2].


Below is a table of [[43-limit]] commas. This table is so big because it can be thought of as two tables interleaved (for the two types of commas) and because these commas are quite large so the more interesting commas appear later. For this reason and completeness, the table shows up to until a little past the largest known lopsided comma that has its own page, the [[Olympia]].
Below is two tables of [[43-limit]] lopsided commas. First, the "top heavy" lopsided commas, where the squared interval is in the numerator, then the "bottom heavy" lopsided commas, where the squared interval is in the denominator. These tables are so big because these commas are quite large so the more interesting commas appear later. For this reason and for completeness, the tables show up to until a little past the largest known lopsided comma that has its own page, which is currently the [[Olympia]].


=== Top-heavy lopsided commas ===
{| class="wikitable center-all
{| class="wikitable center-all
|-
|-
Line 1,186: Line 1,187:
| ([[2/1]])<sup>2</sup> / ([[2/1]])
| ([[2/1]])<sup>2</sup> / ([[2/1]])
| [[2/1]]
| [[2/1]]
|-
| S3<sup>2</sup>*S4 = [[6/5]] * [[9/8]]
| ([[3/2]])<sup>2</sup> / ([[5/3]])
| [[27/20]]
|-
| S4<sup>2</sup>*S5 = [[10/9]] * [[16/15]]
| ([[4/3]])<sup>2</sup> / ([[3/2]])
| [[32/27]]
|-
| S5<sup>2</sup>*S6 = [[15/14]] * [[25/24]]
| ([[5/4]])<sup>2</sup> / ([[7/5]])
| [[125/112]]
|-
| S6<sup>2</sup>*S7 = [[21/20]] * [[36/35]]
| ([[6/5]])<sup>2</sup> / ([[4/3]])
| [[27/25]]
|-
| S7<sup>2</sup>*S8 = [[28/27]] * [[49/48]]
| ([[7/6]])<sup>2</sup> / ([[9/7]])
| [[343/324]]
|-
| S8<sup>2</sup>*S9 = [[36/35]] * [[64/63]]
| ([[8/7]])<sup>2</sup> / ([[5/4]])
| [[256/245]]
|-
| S9<sup>2</sup>*S10 = [[45/44]] * [[81/80]]
| ([[9/8]])<sup>2</sup> / ([[11/9]])
| [[729/704]]
|-
| S10<sup>2</sup>*S11 = [[55/54]] * [[100/99]]
| ([[10/9]])<sup>2</sup> / ([[6/5]])
| [[250/243]]
|-
| S11<sup>2</sup>*S12 = [[66/65]] * [[121/120]]
| ([[11/10]])<sup>2</sup> / ([[13/11]])
| [[1331/1300]]
|-
| S12<sup>2</sup>*S13 = [[78/77]] * [[144/143]]
| ([[12/11]])<sup>2</sup> / ([[7/6]])
| [[864/847]]
|-
| S13<sup>2</sup>*S14 = [[91/90]] * [[169/168]]
| ([[13/12]])<sup>2</sup> / ([[15/13]])
| [[2197/2160]]
|-
| S14<sup>2</sup>*S15 = [[105/104]] * [[196/195]]
| ([[14/13]])<sup>2</sup> / ([[8/7]])
| [[343/338]]
|-
| S15<sup>2</sup>*S16 = [[120/119]] * [[225/224]]
| ([[15/14]])<sup>2</sup> / ([[17/15]])
| [[3375/3332]]
|-
| S16<sup>2</sup>*S17 = [[136/135]] * [[256/255]]
| ([[16/15]])<sup>2</sup> / ([[9/8]])
| [[2048/2025]]
|-
| S17<sup>2</sup>*S18 = [[153/152]] * [[289/288]]
| ([[17/16]])<sup>2</sup> / ([[19/17]])
| [[4913/4864]]
|-
| S18<sup>2</sup>*S19 = [[171/170]] * [[324/323]]
| ([[18/17]])<sup>2</sup> / ([[10/9]])
| [[1458/1445]]
|-
| S19<sup>2</sup>*S20 = [[190/189]] * [[361/360]]
| ([[19/18]])<sup>2</sup> / ([[21/19]])
| [[6859/6804]]
|-
| S20<sup>2</sup>*S21 = [[210/209]] * [[400/399]]
| ([[20/19]])<sup>2</sup> / ([[11/10]])
| [[4000/3971]]
|-
| S21<sup>2</sup>*S22 = [[231/230]] * [[441/440]]
| ([[21/20]])<sup>2</sup> / ([[23/21]])
| [[9261/9200]]
|-
| S22<sup>2</sup>*S23 = [[253/252]] * [[484/483]]
| ([[22/21]])<sup>2</sup> / ([[12/11]])
| [[1331/1323]]
|-
| S23<sup>2</sup>*S24 = [[276/275]] * [[529/528]]
| ([[23/22]])<sup>2</sup> / ([[25/23]])
| [[12167/12100]]
|-
| S24<sup>2</sup>*S25 = [[300/299]] * [[576/575]]
| ([[24/23]])<sup>2</sup> / ([[13/12]])
| [[6912/6877]]
|-
| S25<sup>2</sup>*S26 = [[325/324]] * [[625/624]]
| ([[25/24]])<sup>2</sup> / ([[27/25]])
| [[15625/15552]]
|-
| S26<sup>2</sup>*S27 = [[351/350]] * [[676/675]]
| ([[26/25]])<sup>2</sup> / ([[14/13]])
| [[4394/4375]]
|-
| S27<sup>2</sup>*S28 = [[378/377]] * [[729/728]]
| ([[27/26]])<sup>2</sup> / ([[29/27]])
| [[19683/19604]]
|-
| S28<sup>2</sup>*S29 = [[406/405]] * [[784/783]]
| ([[28/27]])<sup>2</sup> / ([[15/14]])
| [[10976/10935]]
|-
| S29<sup>2</sup>*S30 = [[435/434]] * [[841/840]]
| ([[29/28]])<sup>2</sup> / ([[31/29]])
| [[24389/24304]]
|-
| S30<sup>2</sup>*S31 = [[465/464]] * [[900/899]]
| ([[30/29]])<sup>2</sup> / ([[16/15]])
| [[3375/3364]]
|-
| S31<sup>2</sup>*S32 = [[496/495]] * [[961/960]]
| ([[31/30]])<sup>2</sup> / ([[33/31]])
| [[29791/29700]]
|-
| S32<sup>2</sup>*S33 = [[528/527]] * [[1024/1023]]
| ([[32/31]])<sup>2</sup> / ([[17/16]])
| [[16384/16337]]
|-
| S33<sup>2</sup>*S34 = [[561/560]] * [[1089/1088]]
| ([[33/32]])<sup>2</sup> / ([[35/33]])
| [[35937/35840]]
|-
| S34<sup>2</sup>*S35 = [[595/594]] * [[1156/1155]]
| ([[34/33]])<sup>2</sup> / ([[18/17]])
| [[9826/9801]]
|-
| S35<sup>2</sup>*S36 = [[630/629]] * [[1225/1224]]
| ([[35/34]])<sup>2</sup> / ([[37/35]])
| [[42875/42772]]
|-
| S36<sup>2</sup>*S37 = [[666/665]] * [[1296/1295]]
| ([[36/35]])<sup>2</sup> / ([[19/18]])
| [[23328/23275]]
|-
| S37<sup>2</sup>*S38 = [[703/702]] * [[1369/1368]]
| ([[37/36]])<sup>2</sup> / ([[39/37]])
| [[50653/50544]]
|-
| S38<sup>2</sup>*S39 = [[741/740]] * [[1444/1443]]
| ([[38/37]])<sup>2</sup> / ([[20/19]])
| [[6859/6845]]
|-
| S39<sup>2</sup>*S40 = [[780/779]] * [[1521/1520]]
| ([[39/38]])<sup>2</sup> / ([[41/39]])
| [[59319/59204]]
|-
| S40<sup>2</sup>*S41 = [[820/819]] * [[1600/1599]]
| ([[40/39]])<sup>2</sup> / ([[21/20]])
| [[32000/31941]]
|-
| S41<sup>2</sup>*S42 = [[861/860]] * [[1681/1680]]
| ([[41/40]])<sup>2</sup> / ([[43/41]])
| [[68921/68800]]
|-
| S42<sup>2</sup>*S43 = [[903/902]] * [[1764/1763]]
| ([[42/41]])<sup>2</sup> / ([[22/21]])
| [[18522/18491]]
|-
| S43<sup>2</sup>*S44 = [[946/945]] * [[1849/1848]]
| ([[43/42]])<sup>2</sup> / ([[45/43]])
| [[79507/79380]]
|-
| S44<sup>2</sup>*S45 = [[990/989]] * [[1936/1935]]
| ([[44/43]])<sup>2</sup> / ([[23/22]])
| [[42592/42527]]
|-
| S46<sup>2</sup>*S47 = [[1081/1080]] * [[2116/2115]]
| ([[46/45]])<sup>2</sup> / ([[24/23]])
| [[12167/12150]]
|-
| S49<sup>2</sup>*S50 = [[1225/1224]] * [[2401/2400]]
| ([[49/48]])<sup>2</sup> / ([[51/49]])
| [[117649/117504]]
|-
| S50<sup>2</sup>*S51 = [[1275/1274]] * [[2500/2499]]
| ([[50/49]])<sup>2</sup> / ([[26/25]])
| [[31250/31213]]
|-
| S52<sup>2</sup>*S53 = [[1378/1377]] * [[2704/2703]]
| ([[52/51]])<sup>2</sup> / ([[27/26]])
| [[70304/70227]]
|-
| S55<sup>2</sup>*S56 = [[1540/1539]] * [[3025/3024]]
| ([[55/54]])<sup>2</sup> / ([[57/55]])
| [[166375/166212]]
|-
| S56<sup>2</sup>*S57 = [[1596/1595]] * [[3136/3135]]
| ([[56/55]])<sup>2</sup> / ([[29/28]])
| [[87808/87725]]
|-
| S58<sup>2</sup>*S59 = [[1711/1710]] * [[3364/3363]]
| ([[58/57]])<sup>2</sup> / ([[30/29]])
| [[48778/48735]]
|-
| S63<sup>2</sup>*S64 = [[2016/2015]] * [[3969/3968]]
| ([[63/62]])<sup>2</sup> / ([[65/63]])
| [[250047/249860]]
|-
| S64<sup>2</sup>*S65 = [[2080/2079]] * [[4096/4095]]
| ([[64/63]])<sup>2</sup> / ([[33/32]])
| [[131072/130977]]
|-
| S66<sup>2</sup>*S67 = [[2211/2210]] * [[4356/4355]]
| ([[66/65]])<sup>2</sup> / ([[34/33]])
| [[71874/71825]]
|-
| S70<sup>2</sup>*S71 = [[2485/2484]] * [[4900/4899]]
| ([[70/69]])<sup>2</sup> / ([[36/35]])
| [[42875/42849]]
|-
| S75<sup>2</sup>*S76 = [[2850/2849]] * [[5625/5624]]
| ([[75/74]])<sup>2</sup> / ([[77/75]])
| [[421875/421652]]
|-
| S76<sup>2</sup>*S77 = [[2926/2925]] * [[5776/5775]]
| ([[76/75]])<sup>2</sup> / ([[39/38]])
| [[219488/219375]]
|-
| S78<sup>2</sup>*S79 = [[3081/3080]] * [[6084/6083]]
| ([[78/77]])<sup>2</sup> / ([[40/39]])
| [[59319/59290]]
|}
=== Bottom-heavy lopsided commas ===
{| class="wikitable center-all
|-
! S-expression
! Square Relation
! Comma
|-
|-
| S3<sup>2</sup>*S2 = [[3/2]] * [[9/8]]
| S3<sup>2</sup>*S2 = [[3/2]] * [[9/8]]
| ([[3/1]]) / ([[4/3]])<sup>2</sup>
| ([[3/1]]) / ([[4/3]])<sup>2</sup>
| [[27/16]]
| [[27/16]]
|-
| S3<sup>2</sup>*S4 = [[6/5]] * [[9/8]]
| ([[3/2]])<sup>2</sup> / ([[5/3]])
| [[27/20]]
|-
|-
| S4<sup>2</sup>*S3 = [[6/5]] * [[16/15]]
| S4<sup>2</sup>*S3 = [[6/5]] * [[16/15]]
| ([[2/1]]) / ([[5/4]])<sup>2</sup>
| ([[2/1]]) / ([[5/4]])<sup>2</sup>
| [[32/25]]
| [[32/25]]
|-
| S4<sup>2</sup>*S5 = [[10/9]] * [[16/15]]
| ([[4/3]])<sup>2</sup> / ([[3/2]])
| [[32/27]]
|-
|-
| S5<sup>2</sup>*S4 = [[10/9]] * [[25/24]]
| S5<sup>2</sup>*S4 = [[10/9]] * [[25/24]]
| ([[5/3]]) / ([[6/5]])<sup>2</sup>
| ([[5/3]]) / ([[6/5]])<sup>2</sup>
| [[125/108]]
| [[125/108]]
|-
| S5<sup>2</sup>*S6 = [[15/14]] * [[25/24]]
| ([[5/4]])<sup>2</sup> / ([[7/5]])
| [[125/112]]
|-
|-
| S6<sup>2</sup>*S5 = [[15/14]] * [[36/35]]
| S6<sup>2</sup>*S5 = [[15/14]] * [[36/35]]
| ([[3/2]]) / ([[7/6]])<sup>2</sup>
| ([[3/2]]) / ([[7/6]])<sup>2</sup>
| [[54/49]]
| [[54/49]]
|-
| S6<sup>2</sup>*S7 = [[21/20]] * [[36/35]]
| ([[6/5]])<sup>2</sup> / ([[4/3]])
| [[27/25]]
|-
|-
| S7<sup>2</sup>*S6 = [[21/20]] * [[49/48]]
| S7<sup>2</sup>*S6 = [[21/20]] * [[49/48]]
| ([[7/5]]) / ([[8/7]])<sup>2</sup>
| ([[7/5]]) / ([[8/7]])<sup>2</sup>
| [[343/320]]
| [[343/320]]
|-
| S7<sup>2</sup>*S8 = [[28/27]] * [[49/48]]
| ([[7/6]])<sup>2</sup> / ([[9/7]])
| [[343/324]]
|-
|-
| S8<sup>2</sup>*S7 = [[28/27]] * [[64/63]]
| S8<sup>2</sup>*S7 = [[28/27]] * [[64/63]]
| ([[4/3]]) / ([[9/8]])<sup>2</sup>
| ([[4/3]]) / ([[9/8]])<sup>2</sup>
| [[256/243]]
| [[256/243]]
|-
| S8<sup>2</sup>*S9 = [[36/35]] * [[64/63]]
| ([[8/7]])<sup>2</sup> / ([[5/4]])
| [[256/245]]
|-
|-
| S9<sup>2</sup>*S8 = [[36/35]] * [[81/80]]
| S9<sup>2</sup>*S8 = [[36/35]] * [[81/80]]
| ([[9/7]]) / ([[10/9]])<sup>2</sup>
| ([[9/7]]) / ([[10/9]])<sup>2</sup>
| [[729/700]]
| [[729/700]]
|-
| S9<sup>2</sup>*S10 = [[45/44]] * [[81/80]]
| ([[9/8]])<sup>2</sup> / ([[11/9]])
| [[729/704]]
|-
|-
| S10<sup>2</sup>*S9 = [[45/44]] * [[100/99]]
| S10<sup>2</sup>*S9 = [[45/44]] * [[100/99]]
| ([[5/4]]) / ([[11/10]])<sup>2</sup>
| ([[5/4]]) / ([[11/10]])<sup>2</sup>
| [[125/121]]
| [[125/121]]
|-
| S10<sup>2</sup>*S11 = [[55/54]] * [[100/99]]
| ([[10/9]])<sup>2</sup> / ([[6/5]])
| [[250/243]]
|-
|-
| S11<sup>2</sup>*S10 = [[55/54]] * [[121/120]]
| S11<sup>2</sup>*S10 = [[55/54]] * [[121/120]]
| ([[11/9]]) / ([[12/11]])<sup>2</sup>
| ([[11/9]]) / ([[12/11]])<sup>2</sup>
| [[1331/1296]]
| [[1331/1296]]
|-
| S11<sup>2</sup>*S12 = [[66/65]] * [[121/120]]
| ([[11/10]])<sup>2</sup> / ([[13/11]])
| [[1331/1300]]
|-
|-
| S12<sup>2</sup>*S11 = [[66/65]] * [[144/143]]
| S12<sup>2</sup>*S11 = [[66/65]] * [[144/143]]
| ([[6/5]]) / ([[13/12]])<sup>2</sup>
| ([[6/5]]) / ([[13/12]])<sup>2</sup>
| [[864/845]]
| [[864/845]]
|-
| S12<sup>2</sup>*S13 = [[78/77]] * [[144/143]]
| ([[12/11]])<sup>2</sup> / ([[7/6]])
| [[864/847]]
|-
|-
| S13<sup>2</sup>*S12 = [[78/77]] * [[169/168]]
| S13<sup>2</sup>*S12 = [[78/77]] * [[169/168]]
| ([[13/11]]) / ([[14/13]])<sup>2</sup>
| ([[13/11]]) / ([[14/13]])<sup>2</sup>
| [[2197/2156]]
| [[2197/2156]]
|-
| S13<sup>2</sup>*S14 = [[91/90]] * [[169/168]]
| ([[13/12]])<sup>2</sup> / ([[15/13]])
| [[2197/2160]]
|-
|-
| S14<sup>2</sup>*S13 = [[91/90]] * [[196/195]]
| S14<sup>2</sup>*S13 = [[91/90]] * [[196/195]]
| ([[7/6]]) / ([[15/14]])<sup>2</sup>
| ([[7/6]]) / ([[15/14]])<sup>2</sup>
| [[686/675]]
| [[686/675]]
|-
| S14<sup>2</sup>*S15 = [[105/104]] * [[196/195]]
| ([[14/13]])<sup>2</sup> / ([[8/7]])
| [[343/338]]
|-
|-
| S15<sup>2</sup>*S14 = [[105/104]] * [[225/224]]
| S15<sup>2</sup>*S14 = [[105/104]] * [[225/224]]
| ([[15/13]]) / ([[16/15]])<sup>2</sup>
| ([[15/13]]) / ([[16/15]])<sup>2</sup>
| [[3375/3328]]
| [[3375/3328]]
|-
| S15<sup>2</sup>*S16 = [[120/119]] * [[225/224]]
| ([[15/14]])<sup>2</sup> / ([[17/15]])
| [[3375/3332]]
|-
|-
| S16<sup>2</sup>*S15 = [[120/119]] * [[256/255]]
| S16<sup>2</sup>*S15 = [[120/119]] * [[256/255]]
| ([[8/7]]) / ([[17/16]])<sup>2</sup>
| ([[8/7]]) / ([[17/16]])<sup>2</sup>
| [[2048/2023]]
| [[2048/2023]]
|-
| S16<sup>2</sup>*S17 = [[136/135]] * [[256/255]]
| ([[16/15]])<sup>2</sup> / ([[9/8]])
| [[2048/2025]]
|-
|-
| S17<sup>2</sup>*S16 = [[136/135]] * [[289/288]]
| S17<sup>2</sup>*S16 = [[136/135]] * [[289/288]]
| ([[17/15]]) / ([[18/17]])<sup>2</sup>
| ([[17/15]]) / ([[18/17]])<sup>2</sup>
| [[4913/4860]]
| [[4913/4860]]
|-
| S17<sup>2</sup>*S18 = [[153/152]] * [[289/288]]
| ([[17/16]])<sup>2</sup> / ([[19/17]])
| [[4913/4864]]
|-
|-
| S18<sup>2</sup>*S17 = [[153/152]] * [[324/323]]
| S18<sup>2</sup>*S17 = [[153/152]] * [[324/323]]
| ([[9/8]]) / ([[19/18]])<sup>2</sup>
| ([[9/8]]) / ([[19/18]])<sup>2</sup>
| [[729/722]]
| [[729/722]]
|-
| S18<sup>2</sup>*S19 = [[171/170]] * [[324/323]]
| ([[18/17]])<sup>2</sup> / ([[10/9]])
| [[1458/1445]]
|-
|-
| S19<sup>2</sup>*S18 = [[171/170]] * [[361/360]]
| S19<sup>2</sup>*S18 = [[171/170]] * [[361/360]]
| ([[19/17]]) / ([[20/19]])<sup>2</sup>
| ([[19/17]]) / ([[20/19]])<sup>2</sup>
| [[6859/6800]]
| [[6859/6800]]
|-
| S19<sup>2</sup>*S20 = [[190/189]] * [[361/360]]
| ([[19/18]])<sup>2</sup> / ([[21/19]])
| [[6859/6804]]
|-
|-
| S20<sup>2</sup>*S19 = [[190/189]] * [[400/399]]
| S20<sup>2</sup>*S19 = [[190/189]] * [[400/399]]
| ([[10/9]]) / ([[21/20]])<sup>2</sup>
| ([[10/9]]) / ([[21/20]])<sup>2</sup>
| [[4000/3969]]
| [[4000/3969]]
|-
| S20<sup>2</sup>*S21 = [[210/209]] * [[400/399]]
| ([[20/19]])<sup>2</sup> / ([[11/10]])
| [[4000/3971]]
|-
|-
| S21<sup>2</sup>*S20 = [[210/209]] * [[441/440]]
| S21<sup>2</sup>*S20 = [[210/209]] * [[441/440]]
| ([[21/19]]) / ([[22/21]])<sup>2</sup>
| ([[21/19]]) / ([[22/21]])<sup>2</sup>
| [[9261/9196]]
| [[9261/9196]]
|-
| S21<sup>2</sup>*S22 = [[231/230]] * [[441/440]]
| ([[21/20]])<sup>2</sup> / ([[23/21]])
| [[9261/9200]]
|-
|-
| S22<sup>2</sup>*S21 = [[231/230]] * [[484/483]]
| S22<sup>2</sup>*S21 = [[231/230]] * [[484/483]]
| ([[11/10]]) / ([[23/22]])<sup>2</sup>
| ([[11/10]]) / ([[23/22]])<sup>2</sup>
| [[2662/2645]]
| [[2662/2645]]
|-
| S22<sup>2</sup>*S23 = [[253/252]] * [[484/483]]
| ([[22/21]])<sup>2</sup> / ([[12/11]])
| [[1331/1323]]
|-
|-
| S23<sup>2</sup>*S22 = [[253/252]] * [[529/528]]
| S23<sup>2</sup>*S22 = [[253/252]] * [[529/528]]
| ([[23/21]]) / ([[24/23]])<sup>2</sup>
| ([[23/21]]) / ([[24/23]])<sup>2</sup>
| [[12167/12096]]
| [[12167/12096]]
|-
| S23<sup>2</sup>*S24 = [[276/275]] * [[529/528]]
| ([[23/22]])<sup>2</sup> / ([[25/23]])
| [[12167/12100]]
|-
|-
| S24<sup>2</sup>*S23 = [[276/275]] * [[576/575]]
| S24<sup>2</sup>*S23 = [[276/275]] * [[576/575]]
| ([[12/11]]) / ([[25/24]])<sup>2</sup>
| ([[12/11]]) / ([[25/24]])<sup>2</sup>
| [[6912/6875]]
| [[6912/6875]]
|-
| S24<sup>2</sup>*S25 = [[300/299]] * [[576/575]]
| ([[24/23]])<sup>2</sup> / ([[13/12]])
| [[6912/6877]]
|-
|-
| S25<sup>2</sup>*S24 = [[300/299]] * [[625/624]]
| S25<sup>2</sup>*S24 = [[300/299]] * [[625/624]]
| ([[25/23]]) / ([[26/25]])<sup>2</sup>
| ([[25/23]]) / ([[26/25]])<sup>2</sup>
| [[15625/15548]]
| [[15625/15548]]
|-
| S25<sup>2</sup>*S26 = [[325/324]] * [[625/624]]
| ([[25/24]])<sup>2</sup> / ([[27/25]])
| [[15625/15552]]
|-
|-
| S26<sup>2</sup>*S25 = [[325/324]] * [[676/675]]
| S26<sup>2</sup>*S25 = [[325/324]] * [[676/675]]
| ([[13/12]]) / ([[27/26]])<sup>2</sup>
| ([[13/12]]) / ([[27/26]])<sup>2</sup>
| [[2197/2187]]
| [[2197/2187]]
|-
| S26<sup>2</sup>*S27 = [[351/350]] * [[676/675]]
| ([[26/25]])<sup>2</sup> / ([[14/13]])
| [[4394/4375]]
|-
|-
| S27<sup>2</sup>*S26 = [[351/350]] * [[729/728]]
| S27<sup>2</sup>*S26 = [[351/350]] * [[729/728]]
| ([[27/25]]) / ([[28/27]])<sup>2</sup>
| ([[27/25]]) / ([[28/27]])<sup>2</sup>
| [[19683/19600]]
| [[19683/19600]]
|-
| S27<sup>2</sup>*S28 = [[378/377]] * [[729/728]]
| ([[27/26]])<sup>2</sup> / ([[29/27]])
| [[19683/19604]]
|-
|-
| S28<sup>2</sup>*S27 = [[378/377]] * [[784/783]]
| S28<sup>2</sup>*S27 = [[378/377]] * [[784/783]]
| ([[14/13]]) / ([[29/28]])<sup>2</sup>
| ([[14/13]]) / ([[29/28]])<sup>2</sup>
| [[10976/10933]]
| [[10976/10933]]
|-
| S28<sup>2</sup>*S29 = [[406/405]] * [[784/783]]
| ([[28/27]])<sup>2</sup> / ([[15/14]])
| [[10976/10935]]
|-
|-
| S29<sup>2</sup>*S28 = [[406/405]] * [[841/840]]
| S29<sup>2</sup>*S28 = [[406/405]] * [[841/840]]
| ([[29/27]]) / ([[30/29]])<sup>2</sup>
| ([[29/27]]) / ([[30/29]])<sup>2</sup>
| [[24389/24300]]
| [[24389/24300]]
|-
| S29<sup>2</sup>*S30 = [[435/434]] * [[841/840]]
| ([[29/28]])<sup>2</sup> / ([[31/29]])
| [[24389/24304]]
|-
|-
| S30<sup>2</sup>*S29 = [[435/434]] * [[900/899]]
| S30<sup>2</sup>*S29 = [[435/434]] * [[900/899]]
| ([[15/14]]) / ([[31/30]])<sup>2</sup>
| ([[15/14]]) / ([[31/30]])<sup>2</sup>
| [[6750/6727]]
| [[6750/6727]]
|-
| S30<sup>2</sup>*S31 = [[465/464]] * [[900/899]]
| ([[30/29]])<sup>2</sup> / ([[16/15]])
| [[3375/3364]]
|-
|-
| S31<sup>2</sup>*S30 = [[465/464]] * [[961/960]]
| S31<sup>2</sup>*S30 = [[465/464]] * [[961/960]]
| ([[31/29]]) / ([[32/31]])<sup>2</sup>
| ([[31/29]]) / ([[32/31]])<sup>2</sup>
| [[29791/29696]]
| [[29791/29696]]
|-
| S31<sup>2</sup>*S32 = [[496/495]] * [[961/960]]
| ([[31/30]])<sup>2</sup> / ([[33/31]])
| [[29791/29700]]
|-
|-
| S32<sup>2</sup>*S31 = [[496/495]] * [[1024/1023]]
| S32<sup>2</sup>*S31 = [[496/495]] * [[1024/1023]]
| ([[16/15]]) / ([[33/32]])<sup>2</sup>
| ([[16/15]]) / ([[33/32]])<sup>2</sup>
| [[16384/16335]]
| [[16384/16335]]
|-
| S32<sup>2</sup>*S33 = [[528/527]] * [[1024/1023]]
| ([[32/31]])<sup>2</sup> / ([[17/16]])
| [[16384/16337]]
|-
|-
| S33<sup>2</sup>*S32 = [[528/527]] * [[1089/1088]]
| S33<sup>2</sup>*S32 = [[528/527]] * [[1089/1088]]
| ([[33/31]]) / ([[34/33]])<sup>2</sup>
| ([[33/31]]) / ([[34/33]])<sup>2</sup>
| [[35937/35836]]
| [[35937/35836]]
|-
| S33<sup>2</sup>*S34 = [[561/560]] * [[1089/1088]]
| ([[33/32]])<sup>2</sup> / ([[35/33]])
| [[35937/35840]]
|-
|-
| S34<sup>2</sup>*S33 = [[561/560]] * [[1156/1155]]
| S34<sup>2</sup>*S33 = [[561/560]] * [[1156/1155]]
| ([[17/16]]) / ([[35/34]])<sup>2</sup>
| ([[17/16]]) / ([[35/34]])<sup>2</sup>
| [[4913/4900]]
| [[4913/4900]]
|-
| S34<sup>2</sup>*S35 = [[595/594]] * [[1156/1155]]
| ([[34/33]])<sup>2</sup> / ([[18/17]])
| [[9826/9801]]
|-
|-
| S35<sup>2</sup>*S34 = [[595/594]] * [[1225/1224]]
| S35<sup>2</sup>*S34 = [[595/594]] * [[1225/1224]]
| ([[35/33]]) / ([[36/35]])<sup>2</sup>
| ([[35/33]]) / ([[36/35]])<sup>2</sup>
| [[42875/42768]]
| [[42875/42768]]
|-
| S35<sup>2</sup>*S36 = [[630/629]] * [[1225/1224]]
| ([[35/34]])<sup>2</sup> / ([[37/35]])
| [[42875/42772]]
|-
|-
| S36<sup>2</sup>*S35 = [[630/629]] * [[1296/1295]]
| S36<sup>2</sup>*S35 = [[630/629]] * [[1296/1295]]
| ([[18/17]]) / ([[37/36]])<sup>2</sup>
| ([[18/17]]) / ([[37/36]])<sup>2</sup>
| [[23328/23273]]
| [[23328/23273]]
|-
| S36<sup>2</sup>*S37 = [[666/665]] * [[1296/1295]]
| ([[36/35]])<sup>2</sup> / ([[19/18]])
| [[23328/23275]]
|-
|-
| S37<sup>2</sup>*S36 = [[666/665]] * [[1369/1368]]
| S37<sup>2</sup>*S36 = [[666/665]] * [[1369/1368]]
| ([[37/35]]) / ([[38/37]])<sup>2</sup>
| ([[37/35]]) / ([[38/37]])<sup>2</sup>
| [[50653/50540]]
| [[50653/50540]]
|-
| S37<sup>2</sup>*S38 = [[703/702]] * [[1369/1368]]
| ([[37/36]])<sup>2</sup> / ([[39/37]])
| [[50653/50544]]
|-
|-
| S38<sup>2</sup>*S37 = [[703/702]] * [[1444/1443]]
| S38<sup>2</sup>*S37 = [[703/702]] * [[1444/1443]]
| ([[19/18]]) / ([[39/38]])<sup>2</sup>
| ([[19/18]]) / ([[39/38]])<sup>2</sup>
| [[13718/13689]]
| [[13718/13689]]
|-
| S38<sup>2</sup>*S39 = [[741/740]] * [[1444/1443]]
| ([[38/37]])<sup>2</sup> / ([[20/19]])
| [[6859/6845]]
|-
|-
| S39<sup>2</sup>*S38 = [[741/740]] * [[1521/1520]]
| S39<sup>2</sup>*S38 = [[741/740]] * [[1521/1520]]
| ([[39/37]]) / ([[40/39]])<sup>2</sup>
| ([[39/37]]) / ([[40/39]])<sup>2</sup>
| [[59319/59200]]
| [[59319/59200]]
|-
| S39<sup>2</sup>*S40 = [[780/779]] * [[1521/1520]]
| ([[39/38]])<sup>2</sup> / ([[41/39]])
| [[59319/59204]]
|-
|-
| S40<sup>2</sup>*S39 = [[780/779]] * [[1600/1599]]
| S40<sup>2</sup>*S39 = [[780/779]] * [[1600/1599]]
| ([[20/19]]) / ([[41/40]])<sup>2</sup>
| ([[20/19]]) / ([[41/40]])<sup>2</sup>
| [[32000/31939]]
| [[32000/31939]]
|-
| S40<sup>2</sup>*S41 = [[820/819]] * [[1600/1599]]
| ([[40/39]])<sup>2</sup> / ([[21/20]])
| [[32000/31941]]
|-
|-
| S41<sup>2</sup>*S40 = [[820/819]] * [[1681/1680]]
| S41<sup>2</sup>*S40 = [[820/819]] * [[1681/1680]]
| ([[41/39]]) / ([[42/41]])<sup>2</sup>
| ([[41/39]]) / ([[42/41]])<sup>2</sup>
| [[68921/68796]]
| [[68921/68796]]
|-
| S41<sup>2</sup>*S42 = [[861/860]] * [[1681/1680]]
| ([[41/40]])<sup>2</sup> / ([[43/41]])
| [[68921/68800]]
|-
|-
| S42<sup>2</sup>*S41 = [[861/860]] * [[1764/1763]]
| S42<sup>2</sup>*S41 = [[861/860]] * [[1764/1763]]
| ([[21/20]]) / ([[43/42]])<sup>2</sup>
| ([[21/20]]) / ([[43/42]])<sup>2</sup>
| [[9261/9245]]
| [[9261/9245]]
|-
| S42<sup>2</sup>*S43 = [[903/902]] * [[1764/1763]]
| ([[42/41]])<sup>2</sup> / ([[22/21]])
| [[18522/18491]]
|-
|-
| S43<sup>2</sup>*S42 = [[903/902]] * [[1849/1848]]
| S43<sup>2</sup>*S42 = [[903/902]] * [[1849/1848]]
| ([[43/41]]) / ([[44/43]])<sup>2</sup>
| ([[43/41]]) / ([[44/43]])<sup>2</sup>
| [[79507/79376]]
| [[79507/79376]]
|-
| S43<sup>2</sup>*S44 = [[946/945]] * [[1849/1848]]
| ([[43/42]])<sup>2</sup> / ([[45/43]])
| [[79507/79380]]
|-
|-
| S44<sup>2</sup>*S43 = [[946/945]] * [[1936/1935]]
| S44<sup>2</sup>*S43 = [[946/945]] * [[1936/1935]]
| ([[22/21]]) / ([[45/44]])<sup>2</sup>
| ([[22/21]]) / ([[45/44]])<sup>2</sup>
| [[42592/42525]]
| [[42592/42525]]
|-
| S44<sup>2</sup>*S45 = [[990/989]] * [[1936/1935]]
| ([[44/43]])<sup>2</sup> / ([[23/22]])
| [[42592/42527]]
|-
|-
| S45<sup>2</sup>*S44 = [[990/989]] * [[2025/2024]]
| S45<sup>2</sup>*S44 = [[990/989]] * [[2025/2024]]
| ([[45/43]]) / ([[46/45]])<sup>2</sup>
| ([[45/43]]) / ([[46/45]])<sup>2</sup>
| [[91125/90988]]
| [[91125/90988]]
|-
| S46<sup>2</sup>*S47 = [[1081/1080]] * [[2116/2115]]
| ([[46/45]])<sup>2</sup> / ([[24/23]])
| [[12167/12150]]
|-
|-
| S48<sup>2</sup>*S47 = [[1128/1127]] * [[2304/2303]]
| S48<sup>2</sup>*S47 = [[1128/1127]] * [[2304/2303]]
| ([[24/23]]) / ([[49/48]])<sup>2</sup>
| ([[24/23]]) / ([[49/48]])<sup>2</sup>
| [[55296/55223]]
| [[55296/55223]]
|-
| S49<sup>2</sup>*S50 = [[1225/1224]] * [[2401/2400]]
| ([[49/48]])<sup>2</sup> / ([[51/49]])
| [[117649/117504]]
|-
|-
| S50<sup>2</sup>*S49 = [[1225/1224]] * [[2500/2499]]
| S50<sup>2</sup>*S49 = [[1225/1224]] * [[2500/2499]]
| ([[25/24]]) / ([[51/50]])<sup>2</sup>
| ([[25/24]]) / ([[51/50]])<sup>2</sup>
| [[15625/15606]]
| [[15625/15606]]
|-
| S50<sup>2</sup>*S51 = [[1275/1274]] * [[2500/2499]]
| ([[50/49]])<sup>2</sup> / ([[26/25]])
| [[31250/31213]]
|-
|-
| S51<sup>2</sup>*S50 = [[1275/1274]] * [[2601/2600]]
| S51<sup>2</sup>*S50 = [[1275/1274]] * [[2601/2600]]
| ([[51/49]]) / ([[52/51]])<sup>2</sup>
| ([[51/49]]) / ([[52/51]])<sup>2</sup>
| [[132651/132496]]
| [[132651/132496]]
|-
| S52<sup>2</sup>*S53 = [[1378/1377]] * [[2704/2703]]
| ([[52/51]])<sup>2</sup> / ([[27/26]])
| [[70304/70227]]
|-
|-
| S54<sup>2</sup>*S53 = [[1431/1430]] * [[2916/2915]]
| S54<sup>2</sup>*S53 = [[1431/1430]] * [[2916/2915]]
| ([[27/26]]) / ([[55/54]])<sup>2</sup>
| ([[27/26]]) / ([[55/54]])<sup>2</sup>
| [[39366/39325]]
| [[39366/39325]]
|-
| S55<sup>2</sup>*S56 = [[1540/1539]] * [[3025/3024]]
| ([[55/54]])<sup>2</sup> / ([[57/55]])
| [[166375/166212]]
|-
|-
| S56<sup>2</sup>*S55 = [[1540/1539]] * [[3136/3135]]
| S56<sup>2</sup>*S55 = [[1540/1539]] * [[3136/3135]]
| ([[28/27]]) / ([[57/56]])<sup>2</sup>
| ([[28/27]]) / ([[57/56]])<sup>2</sup>
| [[87808/87723]]
| [[87808/87723]]
|-
| S56<sup>2</sup>*S57 = [[1596/1595]] * [[3136/3135]]
| ([[56/55]])<sup>2</sup> / ([[29/28]])
| [[87808/87725]]
|-
|-
| S57<sup>2</sup>*S56 = [[1596/1595]] * [[3249/3248]]
| S57<sup>2</sup>*S56 = [[1596/1595]] * [[3249/3248]]
| ([[57/55]]) / ([[58/57]])<sup>2</sup>
| ([[57/55]]) / ([[58/57]])<sup>2</sup>
| [[185193/185020]]
| [[185193/185020]]
|-
| S58<sup>2</sup>*S59 = [[1711/1710]] * [[3364/3363]]
| ([[58/57]])<sup>2</sup> / ([[30/29]])
| [[48778/48735]]
|-
|-
| S62<sup>2</sup>*S61 = [[1891/1890]] * [[3844/3843]]
| S62<sup>2</sup>*S61 = [[1891/1890]] * [[3844/3843]]
| ([[31/30]]) / ([[63/62]])<sup>2</sup>
| ([[31/30]]) / ([[63/62]])<sup>2</sup>
| [[59582/59535]]
| [[59582/59535]]
|-
| S63<sup>2</sup>*S64 = [[2016/2015]] * [[3969/3968]]
| ([[63/62]])<sup>2</sup> / ([[65/63]])
| [[250047/249860]]
|-
|-
| S64<sup>2</sup>*S63 = [[2016/2015]] * [[4096/4095]]
| S64<sup>2</sup>*S63 = [[2016/2015]] * [[4096/4095]]
| ([[32/31]]) / ([[65/64]])<sup>2</sup>
| ([[32/31]]) / ([[65/64]])<sup>2</sup>
| [[131072/130975]]
| [[131072/130975]]
|-
| S64<sup>2</sup>*S65 = [[2080/2079]] * [[4096/4095]]
| ([[64/63]])<sup>2</sup> / ([[33/32]])
| [[131072/130977]]
|-
|-
| S65<sup>2</sup>*S64 = [[2080/2079]] * [[4225/4224]]
| S65<sup>2</sup>*S64 = [[2080/2079]] * [[4225/4224]]
| ([[65/63]]) / ([[66/65]])<sup>2</sup>
| ([[65/63]]) / ([[66/65]])<sup>2</sup>
| [[274625/274428]]
| [[274625/274428]]
|-
| S66<sup>2</sup>*S67 = [[2211/2210]] * [[4356/4355]]
| ([[66/65]])<sup>2</sup> / ([[34/33]])
| [[71874/71825]]
|-
|-
| S68<sup>2</sup>*S67 = [[2278/2277]] * [[4624/4623]]
| S68<sup>2</sup>*S67 = [[2278/2277]] * [[4624/4623]]
| ([[34/33]]) / ([[69/68]])<sup>2</sup>
| ([[34/33]]) / ([[69/68]])<sup>2</sup>
| [[157216/157113]]
| [[157216/157113]]
|-
| S70<sup>2</sup>*S71 = [[2485/2484]] * [[4900/4899]]
| ([[70/69]])<sup>2</sup> / ([[36/35]])
| [[42875/42849]]
|-
|-
| S74<sup>2</sup>*S73 = [[2701/2700]] * [[5476/5475]]
| S74<sup>2</sup>*S73 = [[2701/2700]] * [[5476/5475]]
| ([[37/36]]) / ([[75/74]])<sup>2</sup>
| ([[37/36]]) / ([[75/74]])<sup>2</sup>
| [[50653/50625]]
| [[50653/50625]]
|-
| S75<sup>2</sup>*S76 = [[2850/2849]] * [[5625/5624]]
| ([[75/74]])<sup>2</sup> / ([[77/75]])
| [[421875/421652]]
|-
|-
| S76<sup>2</sup>*S75 = [[2850/2849]] * [[5776/5775]]
| S76<sup>2</sup>*S75 = [[2850/2849]] * [[5776/5775]]
| ([[38/37]]) / ([[77/76]])<sup>2</sup>
| ([[38/37]]) / ([[77/76]])<sup>2</sup>
| [[219488/219373]]
| [[219488/219373]]
|-
| S76<sup>2</sup>*S77 = [[2926/2925]] * [[5776/5775]]
| ([[76/75]])<sup>2</sup> / ([[39/38]])
| [[219488/219375]]
|-
|-
| S77<sup>2</sup>*S76 = [[2926/2925]] * [[5929/5928]]
| S77<sup>2</sup>*S76 = [[2926/2925]] * [[5929/5928]]
| ([[77/75]]) / ([[78/77]])<sup>2</sup>
| ([[77/75]]) / ([[78/77]])<sup>2</sup>
| [[456533/456300]]
| [[456533/456300]]
|-
| S78<sup>2</sup>*S79 = [[3081/3080]] * [[6084/6083]]
| ([[78/77]])<sup>2</sup> / ([[40/39]])
| [[59319/59290]]
|-
|-
| S80<sup>2</sup>*S79 = [[3160/3159]] * [[6400/6399]]
| S80<sup>2</sup>*S79 = [[3160/3159]] * [[6400/6399]]
Retrieved from "https://en.xen.wiki/w/S-expression"