{"text_format":"3","workbook":"xlsx","n_sheets":3,"sheets":["Setup","Ratios","Divisors"],"not_rendered":{"images":1,"macros":1,"external_links":1,"defined_names":24,"comments":1},"read":"whole","date_system":"1900"}

{"sheet":"Setup","index":0,"n_rows":37,"n_cols":12,"rows_kept":37}
["Fixed Leap Cycle Finder"]
["by Dr. Irv Bromberg, University of Toronto, Canada"]
["http://individual.utoronto.ca/kalendis/leap/",{"$nulls":3},"Revised 2022.07.11, version 3.33(75)"]
[]
["Remember:  This is Excel, so you can enter valid expressions instead of constants."]
[]
[null,"days","months",{"$nulls":4},"days per week"]
["Length of Short Year:",364,12,{"$f":"\" \"&IF(365.25-SYD>LYD-365.25,\"leap\",\"common\")","v":" common"},{"$nulls":3},7]
["Length of Long Year:",371,12,{"$f":"\" \"&IF(365.25-SYD>LYD-365.25,\"common\",\"leap\")","v":" leap"}]
[{"$nulls":5},"days per month"]
["Calendar Mean Year:","minimum","maximum",null,null,28,{"$nulls":4},false]
["(Time or Fraction >365 days)",{"$f":"B14","v":"05:47:35.929"},{"$f":"C14","v":"05:49:46.962"}]
["Decimal Fractions:",{"$f":"MinFract","v":0.2413880685594416},{"$f":"MaxFract","v":0.24290465005776948},null,{"$f":"\"Report will be output to spreadsheet \"\"\"&ABS(INT(TargetYear))&IF(TargetYear>0,\" AD\",IF(TargetYear<0,\" BC\",))&\" \"&IF(D9=\" leap\",INT(SYD)&\"+\"&INT(B9-B8),INT(LYD)&\"–\"&INT(B9-B8))&\"\"\"\"","v":"Report will be output to spreadsheet \"2022 AD 364+7\""}]
["Suggestions:",{"$f":"MIN(C23:C26)-20/86400","v":"05:47:35.929"},{"$f":"MAX(C23:C26)+15/86400","v":"05:49:46.962"}]
[]
[null,"divisible by","maximum",{"$nulls":6},1]
["Years per Cycle:",1,1000]
[{"$nulls":11},false]
["Target Year (-1000 to 4000):",2022,{"$nulls":9},86400]
[null,{"$f":"IF(TargetYear<-1000,\"Too far in past (limit=-1000)\",IF(TargetYear>4000,\"Too far in future (limit=4000)\",\"\"))","v":""}]
["Color codes for good matches to equinoctial or solstitial mean years (\"Change\" is number of seconds per century, absolute value <1 is reasonably stable):"]
["Mean Year","Solar Longitude","Time >365d","Change","Description",null,"Tighter matches highlighted in boldface:"]
[{"$f":"MeanEqSolstYear(TargetYear,B23)","v":{"v":365.2423697054037,"d":"365.242369705404"}},0,{"$f":"MOD(A23,1)","v":"05:49:00.743"},{"$f":"MeanEqSolstYearChange(TargetYear,B23)","v":{"v":0.1522446982562542,"d":"+0.15"}},"Green = Northward Equinoctial",null,"Green = Northward Equinoctial"]
[{"$f":"MeanEqSolstYear(TargetYear,B24)","v":{"v":365.2416195500409,"d":"365.241619550041"}},{"$f":"B23+90","v":90},{"$f":"MOD(A24,1)","v":"05:47:55.929"},{"$f":"MeanEqSolstYearChange(TargetYear,B24)","v":{"v":-0.6724050734192133,"d":"–0.67"}},"Red = North Solstitial",null,"Red = North Solstitial"]
[{"$f":"MeanEqSolstYear(TargetYear,B25)","v":{"v":365.2420059129945,"d":"365.242005912995"}},{"$f":"B24+90","v":180},{"$f":"MOD(A25,1)","v":"05:48:29.311"},{"$f":"MeanEqSolstYearChange(TargetYear,B25)","v":{"v":-2.73619769141078,"d":"–2.74"}},"Brown = Southward Equinoctial",null,"Brown = Southward Equinoctial"]
[{"$f":"MeanEqSolstYear(TargetYear,B26)","v":{"v":365.24273103894666,"d":"365.242731038947"}},{"$f":"B25+90","v":270},{"$f":"MOD(A26,1)","v":"05:49:31.962"},{"$f":"MeanEqSolstYearChange(TargetYear,B26)","v":{"v":-1.8278001341968775,"d":"–1.83"}},"Blue = South Solstitial",null,"Blue = South Solstitial"]
[]
["Mean Lunar Months:","Days","HH:MM","Seconds","Description"]
["Fixed Mean Month:",{"$f":"29+12/24+44/1440+(2+62/89)/86400","v":29.530586766541823},{"$f":"INT(MOD(B29,1)*1440)/1440","v":"12:44:00"},{"$f":"MOD(86400*MOD(B29,1),60)","v":{"v":2.696629213540291,"d":"2.697"}},"expression in terms of days + hours + minutes + seconds"]
["Hebrew Molad interval:",{"$f":"29+12/24+44/1440+(10/3)/86400","v":29.530594135802467},{"$f":"INT(MOD(B30,1)*1440)/1440","v":"12:44:00"},{"$f":"MOD(86400*MOD(B30,1),60)","v":{"v":3.333333333132032,"d":"3.333"}},"fixed lunar conjunction approximation used for the traditional Hebrew calendar"]
["Synodic Month:",{"$f":"SynodicMonth(TargetYear)","v":29.53058880543702},"12:44:00",{"v":2.8727897585340543,"d":"2.87"},"mean lunar conjunction interval (same ecliptic longitude as Sun)"]
["Saros Period:","  233 synodic months, or about 242 draconic months, or about 239 anomalistic months (used for eclipse prediction)"]
["Sidereal Month:",{"$f":"siderealmonth(TargetYear)","v":27.321661506146874},"07:43:00",{"v":11.554131089946168,"d":"11.55"},"mean interval for Moon to revolve with respect to distant stars"]
["Anomalistic Month:",{"$f":"Anomalisticmonth(TargetYear)","v":27.55455010657511},"13:18:00",{"v":33.129208089536405,"d":"33.13"},"mean interval for Moon to return to its orbital perigee"]
["Draconic (nodical) Month:",{"$f":"DraconicMonth(TargetYear)","v":27.212220732675803},"05:05:00",{"v":35.87130318937852,"d":"35.87"},"mean interval for Moon to return to its orbital ascending node"]
["Eclipse Year:",{"$f":"(Synodic*Draconic)/(Synodic-Draconic)","v":346.6200688238188},"14:52:00",{"v":53.94637794659502,"d":"53.95"},"= (Synodic x Draconic) / (Synodic – Draconic)"]
[{"$nulls":4},"            If 365.25 - DaysPerShortYear > DaysPerLongYear - 365.25 Then"]

{"sheet":"Ratios","index":1,"n_rows":307,"n_cols":8,"rows_kept":307}
["Continued Fraction Approximations (mainly Lunar Months and their Ratios)"]
[]
["Based on continued fraction convergents,"]
["where the coefficients a0, a1, a2, a3 …"]
["correspond to the list of \"alpha\" values."]
[]
[]
[]
[]
[]
[]
["Syntax and how to use the ContinuedFraction( ) macro function:"]
["Select an area of cells that has 5 columns and enough rows for the desired number of convergent rows."]
["Use Ctrl-Shift-Enter to input an expression like the following as a cell ARRAY formula:"]
["{ =ContinuedFraction( TargetValue, FirstDesiredConvergentRow, LastDesiredConvergentRow ) }"]
["The optional convergent row parameters default to rows 2 through 12, although fewer rows usually suffice."]
["You can simply change the first row number below the \"Convergent\" heading to start from the desired row number."]
["If the exact fraction is reached before the last convergent row then the remaining rows are left empty."]
[]
["These columns are returned by the ContinuedFraction( ) function:"]
["alpha shows the improvement in the convergence --- a large value denotes that that convergent is much better than the previous."]
["p is the numerator so far, and q is the denominator so far, and of course p/q is their ratio, which approaches the TargetValue."]
["\"ppt Error\" = \"parts per trillion\" error = (10 ^ 12) * ABS( TargetValue - p/q ) / TargetValue."]
[]
["The cells in the \"Fraction Error\" column are individually calculated by the FractionError( ) function."]
["\"Fraction Error\" = exact fractional difference between target and convergent ratios (where numerator and denominator are integers)"]
[]
[]
["ContinuedFraction( ) can also be called to return a fraction as a string instead of an array of convergents:"]
[]
["ContinuedFraction( TargetValue As Double, Optional MinConvergent As Integer = 2, Optional MaxConvergent As Integer = 12, Optional ErrorLimit As Double = 0.01, Optional JustTheFraction As Boolean ) As String"]
[]
["Uses the continued fraction method to find a series of progressively better fractional approximations (called \"convergents\") to the given"]
["TargetValue. The search terminates when the a convergent's parts-per-trillion error is less than the ErrorLimit or when it reaches the"]
["specified maximum convergent. The final mixed or proper fraction is returned as a text string. If it returns a mixed fraction that you don't"]
["want then pass the inverse of the TargetValue or pass the fractional result to ReduceFractStr( ) with the ImproperFraction flag = True."]
[]
["If the fractional result has an absurd number of digits then try specifying a lower MaxConvergent or a greater ErrorLimit."]
[]
[]
["See examples below."]
[]
[null,"Numerator:",null,"Synodic",{"$f":"Synodic","v":{"v":29.53058880543702,"d":"29.530588805437"}}]
[null,"Denominator:",null,"Sidereal",{"$f":"Sidereal","v":{"v":27.321661506146874,"d":"27.321661506147"}}]
[null,"Ratio:",null,{"$f":"D43&\":\"&D44","v":"Synodic:Sidereal"},{"$f":"Synodic/Sidereal","v":{"v":1.0808489373456727,"d":"1.08084893734567"}}]
[]
["Convergent","alpha","p","q","p/q","ppt Error"]
[4,{"$f":"ContinuedFraction(E45,A48,A60)","v":2},107,99,{"v":1.0808080808080809,"d":"1.08080808080808"},{"v":37800414.267064095,"d":"37800414.27"}]
[{"$f":"A48+1","v":5},2,254,235,{"v":1.0808510638297872,"d":"1.08085106382979"},{"v":1967420.2759500544,"d":"1967420.28"}]
[{"$f":"A49+1","v":6},8,2139,1979,{"v":1.0808489135927235,"d":"1.08084891359272"},{"v":21976.196989615404,"d":"21976.20"}]
[7,10,21644,20025,{"v":1.0808489388264668,"d":"1.08084893882647"},{"v":1370.028782745039,"d":"1370.03"}]
[8,1,23783,22004,{"v":1.0808489365569895,"d":"1.08084893655699"},{"v":729.6885730858145,"d":"729.69"}]
[9,1,45427,42029,{"v":1.0808489376382973,"d":"1.08084893763830"},{"v":270.73588279925883,"d":"270.74"}]
[10,1,69210,64033,{"v":1.0808489372667218,"d":"1.08084893726672"},{"v":73.04521763683724,"d":"73.05"}]
[11,2,183847,170095,{"v":1.080848937358535,"d":"1.08084893735854"},{"v":11.900254869176687,"d":"11.90"}]
[12,2,436904,404223,{"v":1.080848937343991,"d":"1.08084893734399"},{"v":1.555967517377807,"d":"1.56"}]
[13,3,1494559,1382764,1.08084893734578,{"v":0.09922528530413002,"d":"0.10"}]
[14,4,6415140,5935279,{"v":1.080848937345658,"d":"1.08084893734566"},{"v":0.013558734637831431,"d":"0.01"}]
[15,1,7909699,7318043,{"v":1.080848937345681,"d":"1.08084893734568"},{"v":7.806544185418097E-3,"d":"0.01"}]
[16,"","","","",""]
[{"$nulls":5},"exact, within floating point precision limit"]
[]
[null,"Numerator:",null,"Synodic",{"$f":"Synodic","v":{"v":29.53058880543702,"d":"29.530588805437"}}]
[null,"Denominator:",null,"Draconic",{"$f":"Draconic","v":{"v":27.212220732675803,"d":"27.212220732676"}}]
[null,"Ratio:",null,{"$f":"D63&\":\"&D64","v":"Synodic:Draconic"},{"$f":"Synodic/Draconic","v":{"v":1.0851958425420742,"d":"1.08519584254207"}}]
[]
["Convergent","alpha","p","q","p/q","ppt Error"]
[5,{"$f":"ContinuedFraction(E65,A68,A79)","v":4},242,223,{"v":1.0852017937219731,"d":"1.08520179372197"},{"v":5483968.57560471,"d":"5483968.58"}]
[{"$f":"A68+1","v":6},3,777,716,{"v":1.085195530726257,"d":"1.08519553072626"},{"v":287335.9857666164,"d":"287335.99"}]
[{"$f":"A69+1","v":7},5,4127,3803,{"v":1.0851958979752827,"d":"1.08519589797528"},{"v":51081.294541315954,"d":"51081.29"}]
[8,1,4904,4519,{"v":1.0851958397875636,"d":"1.08519583978756"},{"v":2538.2613088006433,"d":"2538.26"}]
[9,16,82591,76107,{"v":1.085195842695153,"d":"1.08519584269515"},{"v":141.0608618807035,"d":"141.06"}]
[10,1,87495,80626,{"v":1.0851958425321857,"d":"1.08519584253219"},{"v":9.112211868198301,"d":"9.11"}]
[11,14,1307521,1204871,{"v":1.0851958425424797,"d":"1.08519584254248"},{"v":0.37362237551825794,"d":"0.37"}]
[12,1,1395016,1285497,{"v":1.085195842541834,"d":"1.08519584254183"},{"v":0.22139069567949346,"d":"0.22"}]
[13,1,2702537,2490368,{"v":1.0851958425421464,"d":"1.08519584254215"},{"v":0.06649905369300867,"d":"0.07"}]
[14,1,4097553,3775865,1.08519584254204,{"v":0.03151032082684103,"d":"0.03"}]
[15,1,6800090,6266233,{"v":1.0851958425420822,"d":"1.08519584254208"},{"v":7.3660490244563444E-3,"d":"0.01"}]
[16,"","","","",""]
[{"$nulls":5},"exact, within floating point precision limit"]
[]
[null,"Numerator:",null,"Synodic",{"$f":"Synodic","v":{"v":29.53058880543702,"d":"29.530588805437"}}]
[null,"Denominator:",null,"Anomalistic",{"$f":"Anomalistic","v":{"v":27.55455010657511,"d":"27.554550106575"}}]
[null,"Ratio:",null,{"$f":"D82&\":\"&D83","v":"Synodic:Anomalistic"},{"$f":"Synodic/Anomalistic","v":{"v":1.0717136985078333,"d":"1.07171369850783"}}]
[]
["Convergent","alpha","p","q","p/q","ppt Error"]
[2,{"$f":"ContinuedFraction(E84,A87,A96)","v":1},15,14,{"v":1.0714285714285714,"d":"1.07142857142857"},{"v":266047807.03919214,"d":"266047807.04"}]
[{"$f":"A87+1","v":3},16,254,237,{"v":1.0717299578059072,"d":"1.07172995780591"},{"v":15171307.501766069,"d":"15171307.50"}]
[{"$f":"A88+1","v":4},1,269,251,{"v":1.0717131474103585,"d":"1.07171314741036"},{"v":514220.79939174594,"d":"514220.80"}]
[5,27,7517,7014,{"v":1.0717137154262903,"d":"1.07171371542629"},{"v":15786.358897246255,"d":"15786.36"}]
[6,1,7786,7265,{"v":1.0717136958017894,"d":"1.07171369580179"},{"v":2524.968969999661,"d":"2524.97"}]
[7,6,54233,50604,{"v":1.071713698521856,"d":"1.07171369852186"},{"v":13.084448723903298,"d":"13.08"}]
[8,27,1472077,1373573,{"v":1.0717136985074691,"d":"1.07171369850747"},{"v":0.33978585193421385,"d":"0.34"}]
[9,1,1526310,1424177,{"v":1.0717136985079805,"d":"1.07171369850798"},{"v":0.1373646462392584,"d":"0.14"}]
[10,2,4524697,4221927,{"v":1.0717136985078142,"d":"1.07171369850781"},{"v":0.017818038577038044,"d":"0.02"}]
[11,2,10575704,9868031,{"v":1.0717136985078382,"d":"1.07171369850784"},{"v":4.55810289180043E-3,"d":"0.00"}]
[{"$nulls":5},"exact, within floating point precision limit"]
[]
[null,"Numerator:",null,30,{"$f":"D99","v":30}]
[null,"Denominator:",null,"Synodic",{"$f":"Synodic","v":{"v":29.53058880543702,"d":"29.530588805437"}}]
[null,"Ratio:",null,{"$f":"D99&\":\"&D100","v":"30:Synodic"},{"$f":"E99/E100","v":{"v":1.0158957614308237,"d":"1.01589576143082"}}]
[]
["Convergent","alpha","p","q","p/q","ppt Error"]
[3,{"$f":"ContinuedFraction(E101,A104,A113)","v":10},703,692,{"v":1.0158959537572254,"d":"1.01589595375723"},{"v":189317.06282792424,"d":"189317.06"}]
[{"$f":"A104+1","v":4},10,7094,6983,{"v":1.0158957468136904,"d":"1.01589574681369"},{"v":14388.418415054437,"d":"14388.42"}]
[{"$f":"A105+1","v":5},1,7797,7675,{"v":1.0158957654723126,"d":"1.01589576547231"},{"v":3978.251583223794,"d":"3978.25"}]
[6,3,30485,30008,{"v":1.0158957611303652,"d":"1.01589576113037"},{"v":295.7572600835592,"d":"295.76"}]
[7,3,99252,97699,{"v":1.0158957614714583,"d":"1.01589576147146"},{"v":39.99879548001987,"d":"40.00"}]
[8,2,228989,225406,{"v":1.015895761426049,"d":"1.01589576142605"},{"v":4.699916390022335,"d":"4.70"}]
[9,3,786219,773917,{"v":1.0158957614317814,"d":"1.01589576143178"},{"v":0.9426935492799298,"d":"0.94"}]
[10,1,1015208,999323,{"v":1.0158957614304884,"d":"1.01589576143049"},{"v":0.330041098866843,"d":"0.33"}]
[11,2,2816635,2772563,{"v":1.0158957614308493,"d":"1.01589576143085"},{"v":0.025135580377276125,"d":"0.03"}]
[12,4,12281748,12089575,{"v":1.0158957614308195,"d":"1.01589576143082"},{"v":0.004152835018854316,"d":"0.00"}]
[{"$nulls":5},"exact, within floating point precision limit"]
[]
[null,"Numerator:",null,30,{"$f":"D116","v":30}]
[null,"Denominator:",null,"Sidereal",{"$f":"Sidereal","v":{"v":27.321661506146874,"d":"27.321661506147"}}]
[null,"Ratio:",null,{"$f":"D116&\":\"&D117","v":"30:Sidereal"},{"$f":"E116/E117","v":{"v":1.0980298541964788,"d":"1.09802985419648"}}]
[]
["Convergent","alpha","p","q","p/q","ppt Error"]
[4,{"$f":"ContinuedFraction(E118,A121,A130)","v":40},2285,2081,{"v":1.0980297933685728,"d":"1.09802979336857"},{"v":55397.315303379735,"d":"55397.32"}]
[{"$f":"A121+1","v":5},3,6911,6294,{"v":1.098029869717191,"d":"1.09802986971719"},{"v":14135.054900044903,"d":"14135.05"}]
[{"$f":"A122+1","v":6},1,9196,8375,{"v":1.0980298507462687,"d":"1.09802985074627"},{"v":3142.182431943751,"d":"3142.18"}]
[7,3,34499,31419,{"v":1.0980298545466118,"d":"1.09802985454661"},{"v":318.87387427521435,"d":"318.87"}]
[8,2,78194,71213,{"v":1.098029854099673,"d":"1.09802985409967"},{"v":88.16326465279177,"d":"88.16"}]
[9,1,112693,102632,{"v":1.0980298542364955,"d":"1.09802985423650"},{"v":36.44405159117262,"d":"36.44"}]
[10,1,190887,173845,{"v":1.098029854180448,"d":"1.09802985418045"},{"v":14.59954138396228,"d":"14.60"}]
[11,1,303580,276477,{"v":1.0980298542012537,"d":"1.09802985420125"},{"v":4.3485586171079404,"d":"4.35"}]
[12,2,798047,726799,{"v":1.0980298541962772,"d":"1.09802985419628"},{"v":0.18361659339350864,"d":"0.18"}]
[13,9,7486003,6817668,{"v":1.0980298541964788,"d":"1.09802985419648"},{"v":0,"d":"0.00"}]
[{"$nulls":5},"exact, within floating point precision limit"]
[]
[null,"Numerator:",null,29.5,{"$f":"D133","v":29.5}]
[null,"Denominator:",null,"Synodic",{"$f":"Synodic","v":{"v":29.53058880543702,"d":"29.530588805437"}}]
[null,"Ratio:",null,{"$f":"D133&\":\"&D134","v":"29.5:Synodic"},{"$f":"E133/E134","v":{"v":0.9989641654069766,"d":"0.998964165406977"}}]
[]
["Convergent","alpha","p","q","p/q","ppt Error"]
[2,{"$f":"ContinuedFraction(E135,A138,A147)","v":964},964,965,{"v":0.9989637305699481,"d":"0.998963730569948"},{"v":435287.9147757212,"d":"435287.91"}]
[{"$f":"A138+1","v":3},2,1929,1931,{"v":0.9989642672190575,"d":"0.998964267219058"},{"v":101917.65070535752,"d":"101917.65"}]
[{"$f":"A139+1","v":4},2,4822,4827,{"v":0.9989641599337062,"d":"0.998964159933706"},{"v":5478.945673926607,"d":"5478.95"}]
[5,7,35683,35720,{"v":0.9989641657334827,"d":"0.998964165733483"},{"v":326.84460136816466,"d":"326.84"}]
[6,2,76188,76267,{"v":0.9989641653664101,"d":"0.998964165366410"},{"v":40.6086139261605,"d":"40.61"}]
[7,3,264247,264521,{"v":0.9989641654159783,"d":"0.998964165415978"},{"v":9.011022212186655,"d":"9.01"}]
[8,1,340435,340788,{"v":0.9989641654048852,"d":"0.998964165404885"},{"v":2.0936067640993117,"d":"2.09"}]
[9,3,1285552,1286885,{"v":0.9989641654071654,"d":"0.998964165407165"},{"v":0.1889336181637557,"d":"0.19"}]
[10,2,2911539,2914558,{"v":0.9989641654068987,"d":"0.998964165406899"},{"v":0.07801847055938618,"d":"0.08"}]
[11,1,4197091,4201443,{"v":0.9989641654069804,"d":"0.998964165406980"},{"v":0.003778672363275114,"d":"0.00"}]
[{"$nulls":5},"exact, within floating point precision limit"]
[]
[null,"Numerator:",null,29.5,{"$f":"D150","v":29.5}]
[null,"Denominator:",null,"Sidereal",{"$f":"Sidereal","v":{"v":27.321661506146874,"d":"27.321661506147"}}]
[null,"Ratio:",null,{"$f":"D150&\":\"&D151","v":"29.5:Sidereal"},{"$f":"E150/E151","v":{"v":1.0797293566265376,"d":"1.07972935662654"}}]
[]
["Convergent","alpha","p","q","p/q","ppt Error"]
[3,{"$f":"ContinuedFraction(E152,A155,A166)","v":1},27,25,{"v":1.08,"d":"1.08000000000000"},{"v":250658530.1228496,"d":"250658530.12"}]
[{"$f":"A155+1","v":4},5,149,138,{"v":1.0797101449275361,"d":"1.07971014492754"},{"v":17793069.053392712,"d":"17793069.05"}]
[{"$f":"A156+1","v":5},2,325,301,{"v":1.079734219269103,"d":"1.07973421926910"},{"v":4503575.396604776,"d":"4503575.40"}]
[6,1,474,439,{"v":1.0797266514806378,"d":"1.07972665148064"},{"v":2505392.562614326,"d":"2505392.56"}]
[7,1,799,740,{"v":1.0797297297297297,"d":"1.07972972972973"},{"v":345552.51260851254,"d":"345552.51"}]
[8,4,3670,3399,{"v":1.0797293321565167,"d":"1.07972933215652"},{"v":22663.106035109977,"d":"22663.11"}]
[9,3,11809,10937,{"v":1.079729359056414,"d":"1.07972935905641"},{"v":2250.449621751478,"d":"2250.45"}]
[10,3,39097,36210,{"v":1.0797293565313448,"d":"1.07972935653134"},{"v":88.16352171328055,"d":"88.16"}]
[11,7,285488,264407,{"v":1.0797293566357926,"d":"1.07972935663579"},{"v":8.571630586016763,"d":"8.57"}]
[12,1,324585,300617,{"v":1.0797293566232116,"d":"1.07972935662321"},{"v":3.0804072490570062,"d":"3.08"}]
[13,2,934658,865641,{"v":1.0797293566270545,"d":"1.07972935662705"},{"v":0.4787494542896529,"d":"0.48"}]
[14,2,2193901,2031899,{"v":1.0797293566264858,"d":"1.07972935662649"},{"v":0.047916075107168866,"d":"0.05"}]
[]
[]
[null,"Numerator:",null,29,{"$f":"D169","v":29}]
[null,"Denominator:",null,"Synodic",{"$f":"Synodic","v":{"v":29.53058880543702,"d":"29.530588805437"}}]
[null,"Ratio:",null,{"$f":"D169&\":\"&D170","v":"29:Synodic"},{"$f":"E169/E170","v":{"v":0.9820325693831295,"d":"0.982032569383130"}}]
[]
["Convergent","alpha","p","q","p/q","ppt Error"]
[2,{"$f":"ContinuedFraction(E171,A174,A182)","v":54},54,55,{"v":0.9818181818181818,"d":"0.981818181818182"},{"v":218310035.3610541,"d":"218310035.36"}]
[{"$f":"A174+1","v":3},1,55,56,{"v":0.9821428571428571,"d":"0.982142857142857"},{"v":112305602.8547234,"d":"112305602.85"}]
[{"$f":"A175+1","v":4},1,109,111,{"v":0.9819819819819819,"d":"0.981981981981982"},{"v":51512956.62157395,"d":"51512956.62"}]
[5,1,164,167,{"v":0.9820359281437125,"d":"0.982035928143713"},{"v":3420213.0231832983,"d":"3420213.02"}]
[6,10,1749,1781,{"v":0.9820325659741718,"d":"0.982032565974172"},{"v":3471.3286391976817,"d":"3471.33"}]
[7,92,161072,164019,{"v":0.9820325693974479,"d":"0.982032569397448"},{"v":14.580294738065431,"d":"14.58"}]
[8,2,323893,329819,{"v":0.9820325693789624,"d":"0.982032569378962"},{"v":4.243353255835152,"d":"4.24"}]
[9,1,484965,493838,0.982032569385102,{"v":2.0085099894273806,"d":"2.01"}]
[10,1,808858,823657,{"v":0.9820325693826435,"d":"0.982032569382643"},{"v":0.4949485947153593,"d":"0.49"}]
[]
[]
[null,"Numerator:",null,29,{"$f":"D185","v":29}]
[null,"Denominator:",null,"Sidereal",{"$f":"Sidereal","v":{"v":27.321661506146874,"d":"27.321661506147"}}]
[null,"Ratio:",null,{"$f":"D185&\":\"&D186","v":"29:Sidereal"},{"$f":"E185/E186","v":{"v":1.0614288590565961,"d":"1.06142885905660"}}]
[]
["Convergent","alpha","p","q","p/q","ppt Error"]
[2,{"$f":"ContinuedFraction(E187,A190,A200)","v":3},52,49,{"v":1.0612244897959184,"d":"1.06122448979592"},192541646.98266092]
[{"$f":"A190+1","v":3},1,69,65,{"v":1.0615384615384615,"d":"1.06153846153846"},103259376.19864431]
[{"$f":"A191+1","v":4},1,121,114,{"v":1.0614035087719298,"d":"1.06140350877193"},23883168.85302806]
[5,2,311,293,{"v":1.0614334470989761,"d":"1.06143344709898"},4322515.202850104]
[6,2,743,700,{"v":1.0614285714285714,"d":"1.06142857142857"},270981.914875275]
[7,6,4769,4493,{"v":1.0614288893834853,"d":"1.06142888938349"},28571.758684858545]
[8,1,5512,5193,{"v":1.0614288465241672,"d":"1.06142884652417"},11807.130327684883]
[9,2,15793,14879,{"v":1.061428859466362,"d":"1.06142885946636"},386.0510983558128]
[10,10,163442,153983,{"v":1.0614288590298928,"d":"1.06142885902989"},25.1578859996545]
[11,1,179235,168862,{"v":1.0614288590683516,"d":"1.06142885906835"},11.075151550325613]
[12,2,521912,491707,{"v":1.061428859056308,"d":"1.06142885905631"},0.27153388070572787]
[]
[]
[null,"Numerator:",null,360000,{"$f":"D203","v":360000}]
[null,"Denominator:",null,354833,{"$f":"D204","v":354833}]
[null,"Ratio:",null,{"$f":"D203&\":\"&D204","v":"360000:354833"},{"$f":"E203/E204","v":{"v":1.0145617797668198,"d":"1.01456177976682"}}]
[]
["Convergent","alpha","p","q","p/q","ppt Error","Fraction Error"]
[2,{"$f":"ContinuedFraction(E205,A208,A218)","v":1},70,69,{"v":1.0144927536231885,"d":"1.01449275362319"},{"v":68035426.73095879,"d":"68035426.73"},{"$f":"FractionError(D$203,D$204,C208,D208)","v":"1690/24483477"}]
[{"$f":"A208+1","v":3},2,209,206,{"v":1.0145631067961165,"d":"1.01456310679612"},{"v":1307982.740071883,"d":"1307982.74"},{"$f":"FractionError(D$203,D$204,C209,D209)","v":"–97/73095598"}]
[{"$f":"A209+1","v":4},17,3623,3571,{"v":1.014561747409689,"d":"1.01456174740969"},{"v":31892.716045340338,"d":"31892.72"},"41/1267108643"]
[5,2,7455,7348,{"v":1.0145617855198694,"d":"1.01456178551987"},{"v":5670.47734992955,"d":"5670.48"},"–15/2607312884"]
[6,2,18533,18267,{"v":1.0145617780697433,"d":"1.01456177806974"},{"v":1672.7187177656422,"d":"1672.72"},"11/6481734411"]
[7,1,25988,25615,1.01456178020691,{"v":433.773669866127,"d":"433.77"},"–4/9089047295"]
[8,2,70509,69497,{"v":1.0145617796451645,"d":"1.01456177964516"},{"v":119.90926021924568,"d":"119.91"},"3/24659829001"]
[9,1,96497,95112,{"v":1.0145617797964506,"d":"1.01456177979645"},{"v":29.205458844537723,"d":"29.21"},"–1/33748876296"]
[10,2,263503,259721,{"v":1.014561779755969,"d":"1.01456177975597"},{"v":10.695135545092507,"d":"10.70"},"1/92157581593"]
[11,1,360000,354833,{"v":1.0145617797668198,"d":"1.01456177976682"},{"v":0,"d":"0.00"},"0","Most accurate solution possible."]
[12,"","","","","",{"$f":"FractionError(D$203,D$204,C218,D218)","v":""},"Although it appears to be exact, the algorithm is"]
[{"$nulls":6},{"$f":"FractionError(D$203,D$204,C219,D219)","v":""},"limited to double-precision floating point."]
[{"$nulls":6},{"$f":"FractionError(D$203,D$204,C220,D220)","v":""},"(Nevertheless, in this case it is exact.)"]
[]
[null,"Numerator:",null,11,{"$f":"D222","v":11}]
[null,"Denominator:",null,117,{"$f":"D223","v":117}]
[null,"Ratio:",null,{"$f":"D222&\":\"&D223","v":"11:117"},{"$f":"E222/E223","v":{"v":0.09401709401709402,"d":"0.094017094017094"}}]
[]
["Convergent","alpha","p","q","p/q","ppt Error","Fraction Error"]
[2,{"$f":"ContinuedFraction(E224,A227,A235)","v":1},1,11,{"v":0.09090909090909091,"d":"0.090909090909091"},{"v":33057851239.669384,"d":"33057851239.67"},{"$f":"FractionError(D$222,D$223,C227,D227)","v":"4/1287"}]
[{"$f":"A227+1","v":3},1,2,21,{"v":0.09523809523809523,"d":"0.095238095238095"},{"v":12987012987.012941,"d":"12987012987.01"},"–1/819"]
[{"$f":"A228+1","v":4},1,3,32,{"v":0.09375,"d":"0.093750000000000"},{"v":2840909090.909081,"d":"2840909090.91"},"1/3744"]
[5,3,11,117,{"v":0.09401709401709402,"d":"0.094017094017094"},{"v":0,"d":"0.00"},"0","Most accurate solution possible."]
[6,"","","","","","","Although it appears to be exact, the algorithm is"]
[7,"","","","","","","limited to double-precision floating point."]
[8,"","","","","",""]
[9,"","","","","",""]
[10,"","","","","",""]
[]
[]
[null,"Numerator:",null,36,{"$f":"D238","v":36}]
[null,"Denominator:",null,117,{"$f":"D239","v":117}]
[null,"Ratio:",null,{"$f":"D238&\":\"&D239","v":"36:117"},{"$f":"E238/E239","v":{"v":0.3076923076923077,"d":"0.307692307692308"}}]
[]
["Convergent","alpha","p","q","p/q","ppt Error","Fraction Error"]
[1,{"$f":"ContinuedFraction(E240,A243,A251)","v":3},1,3,{"v":0.3333333333333333,"d":"0.333333333333333"},{"v":83333333333.3332,"d":"83333333333.33"},{"$f":"FractionError(D$238,D$239,C243,D243)","v":"–1/39"}]
[{"$f":"A243+1","v":2},4,4,13,{"v":0.3076923076923077,"d":"0.307692307692308"},{"v":0,"d":"0.00"},{"$f":"FractionError(D$238,D$239,C244,D244)","v":"0"},"Most accurate solution possible."]
[{"$f":"A244+1","v":3},"","","","","","","36/117 = 4/13"]
[4,"","","","","","","In this case the continued fraction solution"]
[5,"","","","","","","is actually the reduced fraction."]
[6,"","","","","",""]
[7,"","","","","",""]
[8,"","","","","",""]
[9,"","","","","",""]
[]
[]
[null,"Numerator:",null,97,{"$f":"D254","v":97}]
[null,"Denominator:",null,400,{"$f":"D255","v":400}]
[null,"Ratio:",null,{"$f":"D254&\":\"&D255","v":"97:400"},{"$f":"E254/E255","v":{"v":0.2425,"d":"0.242500000000000"}}]
[]
["Convergent","alpha","p","q","p/q","ppt Error","Fraction Error"]
[1,{"$f":"ContinuedFraction(E256,A259,A268)","v":4},1,4,{"v":0.25,"d":"0.250000000000000"},{"v":30927835051.54642,"d":"30927835051.55"},{"$f":"FractionError(D$254,D$255,C259,D259)","v":"–3/400"}]
[{"$f":"A259+1","v":2},8,8,33,{"v":0.24242424242424243,"d":"0.242424242424242"},{"v":312402374.25798917,"d":"312402374.26"},{"$f":"FractionError(D$254,D$255,C260,D260)","v":"1/13200"}]
[{"$f":"A260+1","v":3},12,97,400,{"v":0.2425,"d":"0.242500000000000"},{"v":0,"d":"0.00"},"0"]
[4,"","","","","",""]
[5,"","","","","",""]
[6,"","","","","",""]
[7,"","","","","",""]
[8,"","","","","",""]
[9,"","","","","",""]
[10,"","","","","",""]
[]
[]
[null,"Numerator:",null,473354100,{"$f":"D271","v":473354100},null,{"v":75969.2161365568,"d":"75969.22"},"ppt"]
[null,"Denominator:",null,38271641,{"$f":"D272","v":38271641},null,{"$f":"G271/1000","v":75.96921613655681},"ppb"]
[null,"Ratio:",null,{"$f":"D271&\":\"&D272","v":"473354100:38271641"},{"$f":"E271/E272","v":{"v":12.368272894282217,"d":"12.368272894282200"}},null,{"$f":"G272/1000","v":0.07596921613655681},"ppm"]
[]
["Convergent","alpha","p","q","p/q","ppt Error","Fraction Error"]
[1,{"$f":"ContinuedFraction(E273,A276,A285)","v":2},25,2,{"v":12.5,"d":"12.500000000000000"},{"v":10650404211.139242,"d":"10650404211.14"},{"$f":"FractionError(D$271,D$272,C276,D276)","v":"–10082825/76543282"}]
[{"$f":"A276+1","v":2},1,37,3,{"v":12.333333333333334,"d":"12.333333333333300"},{"v":2824934511.6758995,"d":"2824934511.68"},{"$f":"FractionError(D$271,D$272,C277,D277)","v":"4011583/114814923"}]
[{"$f":"A277+1","v":3},2,99,8,{"v":12.375,"d":"12.375000000000000"},{"v":543900169.0278504,"d":"543900169.03"},"–2059659/306173128"]
[4,1,136,11,{"v":12.363636363636363,"d":"12.363636363636400"},{"v":374872925.7095751,"d":"374872925.71"},"1951924/420988051"]
[5,1,235,19,{"v":12.368421052631579,"d":"12.368421052631600"},{"v":11978903.653551402,"d":"11978903.65"},"–107735/727161179"]
[6,18,4366,353,{"v":12.36827195467422,"d":"12.368271954674200"},{"v":75969.21613655679,"d":"75969.22"},"12694/13509889273"]
[7,8,35163,2843,{"v":12.368272951107985,"d":"12.368272951108000"},{"v":4594.478767457106,"d":"4594.48"},"–6183/108806275363"]
[8,2,74692,6039,{"v":12.368272892863057,"d":"12.368272892863100"},{"v":114.74196218312422,"d":"114.74"},"328/231122439999"]
[9,18,1379619,111545,{"v":12.368272894347573,"d":"12.368272894347600"},{"v":5.284142854369554,"d":"5.28"},"–279/4269010195345"]
[10,1,1454311,117584,{"v":12.36827289427133,"d":"12.368272894271300"},{"v":0.8802596095463958,"d":"0.88"},"49/4500132635344"]
[]
[]
["Same ratio, but return only specifed convergent as a fraction, and use ReduceFractStr( ) to convert to improper fraction:"]
["Convergent","Mixed","Improper"]
[5,{"$f":"ContinuedFraction(E273,2,A290,,TRUE)","v":"12+7/19"},{"$f":"ReduceFractStr(B290,TRUE)","v":"235/19"},null,"Mixed shows the number of leap months per cycle."]
[6,{"$f":"ContinuedFraction(E273,3,A291,,TRUE)","v":"12+130/353"},{"$f":"ReduceFractStr(B291,TRUE)","v":"4366/353"},null,"Improper shows the total number of months per cycle."]
{"$rows_empty":7}
[{"$nulls":3},"Fractional part of Tibetan mean synodic month:",{"v":0.5305869872701556,"d":"0.530586987270156"}]
[]
["Convergent","alpha","p","q","p/q","ppt Error"]
[5,{"$f":"ContinuedFraction(E299,A302,A307)","v":2},26,49,0.5306122448979592,{"v":47603179.89239977,"d":"47603179.89"}]
[{"$f":"A302+1","v":6},16,425,801,0.5305867665418227,{"v":416007.81444031885,"d":"416007.81"}]
[{"$f":"A303+1","v":7},6,2576,4855,0.5305870236869207,{"v":68634.86274949087,"d":"68634.86"}]
[8,1,3001,5656,0.5305869872701556,{"v":0,"d":"0.00"},"exact"]
[9,"","","","",""]
[10,"","","","",""]

{"sheet":"Divisors","index":2,"n_rows":24,"n_cols":4,"rows_kept":24}
["Factorization of Integers"]
[]
["It may be helpful to find the integer divisors of the number of years per leap cycle, number of days per cycle, etc."]
[]
["For that purpose, this spreadsheet include a macro Divisors( ) function."]
[]
["Syntax and how to use the Divisors macro function:"]
[]
["=Divisors( TheNumber {, FlagPrimes } )"]
[]
["The values for the optional FlagPrimes parameter are:"]
[1,"Return all Divisors, with flagged primes (default)."]
[0,"Return all Divisors without flagging primes"]
[-1,"Return primes divisors only (factorize) and flag all as primes."]
[-2,"(or any value <-1) Return prime divisors only (factorize) but don't flag as primes."]
[]
["See examples below:"]
[]
["Number","FlagPrimes","Syntax","Result"]
[3150,0,{"$f":"\"=Divisors(\"&A20&\",\"&B20&\")\"","v":"=Divisors(3150,0)"},{"$f":"divisors(A20,B20)","v":"2, 3, 5, 6, 7, 9, 10, 14, 15, 18, 21, 25, 30, 35, 42, 45, 50, 63, 70, 75, 90, 105, 126, 150, 175, 210, 225, 315, 350, 450, 525, 630, 1050, 1575, 3150"}]
[3150,1,{"$f":"\"=Divisors(\"&A21&\",\"&B21&\")\"","v":"=Divisors(3150,1)"},{"$f":"divisors(A21,B21)","v":"2', 3', 5', 6, 7', 9, 10, 14, 15, 18, 21, 25, 30, 35, 42, 45, 50, 63, 70, 75, 90, 105, 126, 150, 175, 210, 225, 315, 350, 450, 525, 630, 1050, 1575, 3150"}]
[3150,-1,{"$f":"\"=Divisors(\"&A22&\",\"&B22&\")\"","v":"=Divisors(3150,-1)"},{"$f":"divisors(A22,B22)","v":"2', 3', 5', 7', 3150"}]
[3150,-2,{"$f":"\"=Divisors(\"&A23&\",\"&B23&\")\"","v":"=Divisors(3150,-2)"},{"$f":"divisors(A23,B23)","v":"2, 3, 5, 7, 3150"}]
[293,1,{"$f":"\"=Divisors(\"&A24&\",\"&B24&\")\"","v":"=Divisors(293,1)"},{"$f":"divisors(A24,B24)","v":"293'"}]