A table of the ratios between the Expected Duration of the Dym-Luks Ball and\ Cell game with n cells and r balls and r / n \(j - 1) ----- |-----| \ \n - 1/ ) -------------- / j ----- j = 1 and the ratios between the variance and / r / n \(2 j - 2)\ / r / n \(j - 1)\ |----- |-----| | |----- |-----| | | \ \n - 1/ | | \ \n - 1/ | | ) ----------------| - | ) --------------| | / 2 | | / j | |----- j | |----- | \j = 1 / \j = 1 / for n from 2 to, 10, and r from 1 to, 50 By Shalosh B. Ekhad Let's call the ratios for the expectation RatE_n(r) Let's call the ratios for the variance RatE2_n(r) when there are, 2, cells, the first , 50, starting at r=3, terms of the seqence RatE_n(r) are [1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1. , 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1\ ., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.] Also the terms of terms of the seqence RatE2_n(r) are [5.500000000, 1.818181818, 1.210871603, 1.061358058, 1.018683812, 1.005762949, 1.001773543, 1.000540844, 1.000162989, 1.000048512, 1.000014267, 1.000004149, 1\ .000001195, 1.000000341, 1.000000097, 1.000000027, 1.000000008, 1.000000002, 1.\ 000000001, 1.000000000, 1.000000000, 1.000000000, 1.000000000, 1.000000000, 1.0\ 00000000, 1.000000000, 1.000000000, 1.000000000, 1.000000000, 1.000000000, 1.00\ 0000000, 1.000000000, 1.000000000, 1.000000000, 1.000000000, 1.000000000, 1.000\ 000000, 1.000000000, 1.000000000, 1.000000000, 1.000000000, 1.000000000, 1.0000\ 00000, 1.000000000, 1.000000000, 1.000000000, 1.000000000, 1.000000000, 1.00000\ 0000, 1.000000000] when there are, 3, cells, the first , 50, starting at r=3, terms of the seqence RatE_n(r) are [.9000000000, .9420560748, .9641319943, .9793094929, .9885444435, .9940809141, .9972637148, .9990027005, .9998865335, 1.000284755, 1.000421690, 1.000428502, 1\ .000378562, 1.000310626, 1.000243532, 1.000185165, 1.000137752, 1.000100845, 1.\ 000072930, 1.000052246, 1.000037151, 1.000026259, 1.000018471, 1.000012941, 1.0\ 00009036, 1.000006292, 1.000004371, 1.000003030, 1.000002096, 1.000001448, 1.00\ 0000999, 1.000000688, 1.000000474, 1.000000326, 1.000000224, 1.000000153, 1.000\ 000105, 1.000000072, 1.000000049, 1.000000034, 1.000000023, 1.000000016, 1.0000\ 00011, 1.000000007, 1.000000005, 1.000000003, 1.000000002, 1.000000002, 1.00000\ 0001, 1.000000001] Also the terms of terms of the seqence RatE2_n(r) are [-3., -2.974335260, -4.112876452, -18.04855097, 5.046604308, 2.208432606, 1.522\ 317400, 1.255193771, 1.131774415, 1.069932831, 1.037613026, 1.020344365, 1.0110\ 16731, 1.005956903, 1.003211347, 1.001724622, 1.000922303, 1.000491111, 1.00026\ 0402, 1.000137513, 1.000072341, 1.000037921, 1.000019813, 1.000010321, 1.000005\ 362, 1.000002779, 1.000001437, 1.000000741, 1.000000382, 1.000000196, 1.0000001\ 01, 1.000000052, 1.000000026, 1.000000014, 1.000000007, 1.000000004, 1.00000000\ 2, 1.000000001, 1.000000000, 1.000000000, 1.000000000, 1.000000000, 1.000000000 , 1.000000000, 1.000000000, 1.000000000, 1.000000000, 1.000000000, 1.000000000, 1.000000000] when there are, 4, cells, the first , 50, starting at r=3, terms of the seqence RatE_n(r) are [.8262295082, .8727272727, .9048901488, .9286852606, .9484844538, .9634212802, .9749122380, .9835064890, .9898184444, .9943377004, .9974756979, .9995698096, 1\ .000892433, 1.001659384, 1.002038164, 1.002155903, 1.002106766, 1.001958638, 1.\ 001758965, 1.001539696, 1.001321338, 1.001116183, 1.000930806, 1.000767948, 1.0\ 00627893, 1.000509447, 1.000410620, 1.000329073, 1.000262405, 1.000208327, 1.00\ 0164755, 1.000129850, 1.000102028, 1.000079948, 1.000062494, 1.000048743, 1.000\ 037942, 1.000029482, 1.000022870, 1.000017715, 1.000013703, 1.000010587, 1.0000\ 08170, 1.000006298, 1.000004850, 1.000003732, 1.000002869, 1.000002204, 1.00000\ 1692, 1.000001298] Also the terms of terms of the seqence RatE2_n(r) are [-1.648757396, -1.341011673, -1.231781734, -1.263989057, -1.369957509, -1.66308\ 5599, -2.385811985, -5.027970641, 29.78806576, 3.953312311, 2.262382517, 1.6836\ 96586, 1.408589309, 1.257220251, 1.166995200, 1.110544061, 1.074108427, 1.05010\ 1162, 1.034060679, 1.023240915, 1.015895425, 1.010886973, 1.007462338, 1.005116\ 541, 1.003508111, 1.002404753, 1.001647799, 1.001128584, 1.000772565, 1.0005285\ 60, 1.000361416, 1.000246988, 1.000168695, 1.000115158, 1.000078571, 1.00005358\ 2, 1.000036523, 1.000024884, 1.000016946, 1.000011536, 1.000007850, 1.000005340 , 1.000003631, 1.000002468, 1.000001677, 1.000001139, 1.000000774, 1.000000525, 1.000000356, 1.000000242] when there are, 5, cells, the first , 50, starting at r=3, terms of the seqence RatE_n(r) are [.7766990291, .8191511899, .8536141501, .8789655192, .9011967460, .9206665420, .9367380847, .9503640586, .9616951458, .9710574671, .9787053857, .9848767303, .\ 9897863776, .9936288272, .9965786657, .9987911236, 1.000402669, 1.001531790, 1.\ 002279992, 1.002733018, 1.002962232, 1.003026123, 1.002971852, 1.002836795, 1.0\ 02650020, 1.002433666, 1.002204188, 1.001973463, 1.001749741, 1.001538460, 1.00\ 1342911, 1.001164794, 1.001004651, 1.000862218, 1.000736689, 1.000626923, 1.000\ 531588, 1.000449278, 1.000378585, 1.000318151, 1.000266700, 1.000223061, 1.0001\ 86171, 1.000155082, 1.000128953, 1.000107050, 1.000088730, 1.000073440, 1.00006\ 0704, 1.000050115] Also the terms of terms of the seqence RatE2_n(r) are [-1.219730942, -.9961554484, -.8520716065, -.8098653990, -.7953331177, -.795058\ 4337, -.8304416811, -.9014135248, -1.030281015, -1.258929511, -1.692580530, -2.\ 675392628, -6.347573709, 21.88628747, 4.393971547, 2.603950036, 1.943886684, 1.\ 611506534, 1.417777831, 1.294998028, 1.212934433, 1.156077477, 1.115673950, 1.0\ 86429753, 1.064972373, 1.049066271, 1.037183067, 1.028252217, 1.021509381, 1.01\ 6400515, 1.012519149, 1.009564223, 1.007311049, 1.005590932, 1.004276610, 1.003\ 271716, 1.002503068, 1.001914960, 1.001464913, 1.001120497, 1.000856922, 1.0006\ 55228, 1.000500905, 1.000382847, 1.000292547, 1.000223493, 1.000170698, 1.00013\ 0343, 1.000099504, 1.000075944] when there are, 6, cells, the first , 50, starting at r=3, terms of the seqence RatE_n(r) are [.7417582418, .7779799818, .8122678462, .8377316485, .8593887144, .8796879112, .8976938595, .9133376375, .9272361541, .9394269583, .9500739563, .9593312617, .\ 9673249319, .9741796419, .9800109068, .9849279204, .9890334023, .9924236617, .9\ 951884715, .9974109153, .9991672602, 1.000526895, 1.001552359, 1.002299465, 1.0\ 02817520, 1.003149637, 1.003333113, 1.003399863, 1.003376891, 1.003286781, 1.00\ 3148191, 1.002976331, 1.002783423, 1.002579131, 1.002370954, 1.002164581, 1.001\ 964208, 1.001772810, 1.001592384, 1.001424150, 1.001268723, 1.001126255, 1.0009\ 96556, 1.000879186, 1.000773532, 1.000678872, 1.000594417, 1.000519355, 1.00045\ 2868, 1.000394161] Also the terms of terms of the seqence RatE2_n(r) are [-.9903961585, -.8388778709, -.7003810097, -.6386285281, -.6158371079, -.595570\ 2204, -.5837573607, -.5870449199, -.6008395254, -.6287261113, -.6751068705, -.7\ 466759738, -.8556238156, -1.024038061, -1.296816880, -1.780585775, -2.804918429 , -6.142652636, 68.29186391, 5.779787306, 3.211477498, 2.324543408, 1.883919428 , 1.625751285, 1.459578817, 1.346002635, 1.265117599, 1.205789582, 1.161310448, 1.127404048, 1.101221652, 1.080797567, 1.064736307, 1.052023893, 1.041909318, 1\ .033827459, 1.027347420, 1.022137018, 1.017937802, 1.014547135, 1.011805111, 1.\ 009584863, 1.007785276, 1.006325459, 1.005140492, 1.004178133, 1.003396254, 1.0\ 02760818, 1.002244285, 1.001824344] when there are, 7, cells, the first , 50, starting at r=3, terms of the seqence RatE_n(r) are [.7159090909, .7456015216, .7785892815, .8039374141, .8245325914, .8439737674, .8623238227, .8787446473, .8934935954, .9069593475, .9191393447, .9301127620, .\ 9399918844, .9488466165, .9567514825, .9637769344, .9699894332, .9754531460, .9\ 802297612, .9843786730, .9879569381, .9910191752, .9936174233, .9958009870, .99\ 76162891, .9991067464, 1.000312679, 1.001271257, 1.002016494, 1.002579278, 1.00\ 2987443, 1.003265877, 1.003436660, 1.003519223, 1.003530531, 1.003485268, 1.003\ 396040, 1.003273567, 1.003126882, 1.002963517, 1.002789685, 1.002610442, 1.0024\ 29847, 1.002251101, 1.002076677, 1.001908434, 1.001747714, 1.001595440, 1.00145\ 2186, 1.001318246] Also the terms of terms of the seqence RatE2_n(r) are [-.8416870416, -.7430139144, -.6197395029, -.5476347011, -.5201378106, -.501962\ 2913, -.4827306657, -.4706433758, -.4667256664, -.4674636277, -.4739560734, -.4\ 875966219, -.5093206035, -.5412398280, -.5863838673, -.6492959540, -.7371227332 , -.8616120837, -1.043483790, -1.322946906, -1.789621606, -2.692102234, -5.0715\ 46324, -26.33201052, 9.505442092, 4.304848846, 2.907899566, 2.267534266, 1.9050\ 62180, 1.675047417, 1.518291251, 1.406165230, 1.323135437, 1.260045898, 1.21115\ 2202, 1.172670290, 1.142008274, 1.117333421, 1.097315218, 1.080966234, 1.067539\ 899, 1.056462777, 1.047288431, 1.039665264, 1.033313608, 1.028009094, 1.0235703\ 88, 1.019849994, 1.016727280, 1.014103109] when there are, 8, cells, the first , 50, starting at r=3, terms of the seqence RatE_n(r) are [.6960451977, .7195317817, .7506320939, .7757760663, .7955466969, .8136582375, .8313671481, .8479107755, .8628829816, .8766451774, .8894388507, .9012366315, .\ 9120767283, .9220474414, .9311952711, .9395634885, .9471993572, .9541450895, .9\ 604415710, .9661285989, .9712448044, .9758279040, .9799147155, .9835411473, .98\ 67421442, .9895516047, .9920022822, .9941256812, .9959519544, .9975098085, .998\ 8264232, .9999273867, 1.000836649, 1.001576497, 1.002167546, 1.002628753, 1.002\ 977447, 1.003229373, 1.003398751, 1.003498348, 1.003539554, 1.003532468, 1.0034\ 85991, 1.003407913, 1.003305007, 1.003183120, 1.003047261, 1.002901689, 1.00274\ 9989, 1.002595153] Also the terms of terms of the seqence RatE2_n(r) are [-.7352361792, -.6748943767, -.5694022756, -.4926796615, -.4588330532, -.442966\ 8247, -.4260635279, -.4094857535, -.3989718998, -.3933683396, -.3901677289, -.3\ 899472665, -.3935141570, -.4008922965, -.4125960708, -.4294948598, -.4526325396 , -.4834902157, -.5241349519, -.5774951918, -.6478550790, -.7417154449, -.86942\ 63944, -1.048521097, -1.311251805, -1.724146554, -2.450048811, -4.021394294, -9\ .764700358, 32.94927282, 6.675511245, 3.898873327, 2.847597236, 2.299994135, 1.\ 967346171, 1.746070826, 1.589827401, 1.474774753, 1.387394287, 1.319449406, 1.2\ 65636107, 1.222385406, 1.187208497, 1.158317816, 1.134397264, 1.114457288, 1.09\ 7740357, 1.083657504, 1.071744616, 1.061631663] when there are, 9, cells, the first , 50, starting at r=3, terms of the seqence RatE_n(r) are [.6803149606, .6981089847, .7270191408, .7518411594, .7711083983, .7879878479, .8045791763, .8207232976, .8356752978, .8493827404, .8621984878, .8742649754, .\ 8855373078, .8960340063, .9058278058, .9149565346, .9234439010, .9313226358, .9\ 386227750, .9453710789, .9515942314, .9573181352, .9625680366, .9673687753, .97\ 17448021, .9757202132, .9793187479, .9825637610, .9854781789, .9880844431, .990\ 4044463, .9924594660, .9942700971, .9958561887, .9972367859, .9984300788, .9994\ 533605, 1.000322994, 1.001054389, 1.001661989, 1.002159268, 1.002558738, 1.0028\ 71958, 1.003109561, 1.003281280, 1.003395983, 1.003461712, 1.003485722, 1.00347\ 4529, 1.003433957] Also the terms of terms of the seqence RatE2_n(r) are [-.6543497758, -.6220362600, -.5340992112, -.4570260420, -.4167660021, -.400316\ 2223, -.3873059211, -.3715659612, -.3578067557, -.3490092572, -.3432758320, -.3\ 388333402, -.3361612982, -.3358729849, -.3378201300, -.3420419350, -.3489151884 , -.3588023024, -.3721484900, -.3895793429, -.4118990943, -.4401614609, -.47577\ 23353, -.5206362121, -.5773934074, -.6498080244, -.7434330421, -.8668143153, -1\ .033821170, -1.268559843, -1.616964469, -2.178751796, -3.218757295, -5.75026708\ 0, -20.54585771, 15.31897346, 5.938046485, 3.826869438, 2.901231146, 2.38519353\ 8, 2.058573522, 1.834938636, 1.673436346, 1.552247449, 1.458657585, 1.384756753 , 1.325366122, 1.276953268, 1.237026479, 1.203778230] when there are, 10, cells, the first , 50, starting at r=3, terms of the seqence RatE_n(r) are [.6675542031, .6801986612, .7067841168, .7311450269, .7501418462, .7660790167, .7814474632, .7968109661, .8115144174, .8250978018, .8377156985, .8496674753, .\ 8610250210, .8717323489, .8817982934, .8912856409, .9002293480, .9086418114, .9\ 165441215, .9239602958, .9309093028, .9374092047, .9434782932, .9491341113, .95\ 43938047, .9592743234, .9637924144, .9679646740, .9718075687, .9753374351, .978\ 5704674, .9815226945, .9842099482, .9866478255, .9888516476, .9908364165, .9926\ 167714, .9942069464, .9956207309, .9968714329, .9979718463, .9989342236, .99977\ 02532, 1.000491042, 1.001107103, 1.001628350, 1.002064092, 1.002423044, 1.00271\ 3325, 1.002942475] Also the terms of terms of the seqence RatE2_n(r) are [-.5903796845, -.5787888438, -.5070378843, -.4325582117, -.3870521410, -.367781\ 2587, -.3573007801, -.3446429687, -.3304823048, -.3192703863, -.3119706599, -.3\ 065627735, -.3018242377, -.2982378606, -.2963280638, -.2959286130, -.2968916537 , -.2994048233, -.3036681403, -.3098299758, -.3181220483, -.3288549689, -.34240\ 10664, -.3592284896, -.3799296481, -.4052563144, -.4361756087, -.4739501548, -.\ 5202577595, -.5773757268, -.6484737541, -.7380981289, -.8530105432, -1.00372638\ 8, -1.207542567, -1.495060940, -1.926037054, -2.634969948, -4.001176763, -7.671\ 106285, -48.65817727, 12.51836889, 5.833640062, 3.927813291, 3.030526434, 2.511\ 662308, 2.175466808, 1.941288064, 1.769818191, 1.639609755] This ends this paper that took, 229.083, seconds to generate .