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My favorite integer sequence
#51
Just an accidental finding.

[update]: Hmm, after some more investigation it seems, that the halfiterative of the sinh alone is not that "nugget" which I felt it were in the beginning. Still it might be a good starting point, but after some regressions with various cofactors it seems to me, that likely it cannot substantially be improved by adding terms of standardfunctions and/or further fractional iterates. So I think, Jay's very well worked function is still the way to go and this posting should only survive for historical reasons... [/update]



I use the sequence A[k]=(1,1,2,2,4,4,6,6,...) with k beginning at 0 and tried the half-iterate of the asinh()-function to translate A[k]-> k^ .
I get that nice approximation:

Code:
.  
  k  |  k^ = -1            |      |
     |  +2*asinh°0.5(A[k]) | A[k] |  d = k^ - k
-----+---------------------+------+-------------------
   0   0.871122566717836     1     0.8711225667178365
   1   0.871122566717836     1    -0.1288774332821635
   2   2.333123406391677     2     0.3331234063916770
   3   2.333123406391677     2    -0.6668765936083230
   4   4.444125935755581     4     0.4441259357555807
   5   4.444125935755581     4    -0.5558740642444193
   6   5.996686817157767     6  -0.003313182842233339
   7   5.996686817157767     6     -1.003313182842233
   8   8.309891291850146    10     0.3098912918501464
   9   8.309891291850146    10    -0.6901087081498536
  10   10.06605408134559    14    0.06605408134559265
  11   10.06605408134559    14    -0.9339459186544073
  12   12.14359799584454    20     0.1435979958445373
  13   12.14359799584454    20    -0.8564020041554627
  14   13.82177298614942    26    -0.1782270138505756
  15   13.82177298614942    26     -1.178227013850576
  16   16.08985351232628    36    0.08985351232627991
  17   16.08985351232628    36    -0.9101464876737201
  18   17.94134162168168    46   -0.05865837831832136
  19   17.94134162168168    46     -1.058658378318321
  20   20.09383140688473    60    0.09383140688473422
  21   20.09383140688473    60    -0.9061685931152658
  22   21.90407738775139    74   -0.09592261224861381
  23   21.90407738775139    74     -1.095922612248614
  24   24.09366985432521    94    0.09366985432520982
  25   24.09366985432521    94    -0.9063301456747902
  26   25.95938189509372   114   -0.04061810490628340
  27   25.95938189509372   114     -1.040618104906283
  28   28.04805382734139   140    0.04805382734139237
  29   28.04805382734139   140    -0.9519461726586076
  30   29.86181858151267   166    -0.1381814184873316
  31   29.86181858151267   166     -1.138181418487332
  32   32.04658019188079   202    0.04658019188078527
  33   32.04658019188079   202    -0.9534198081192147
  34   33.95209498128483   238   -0.04790501871516933
  35   33.95209498128483   238     -1.047905018715169
  36   36.08894669829039   284    0.08894669829039405
  37   36.08894669829039   284    -0.9110533017096059
  38   37.97426693042275   330   -0.02573306957725277
  39   37.97426693042275   330     -1.025733069577253
  40   40.14969749681103   390     0.1496974968110288
  41   40.14969749681103   390    -0.8503025031889712
  42   42.07952641659918   450    0.07952641659918464
  43   42.07952641659918   450    -0.9204735834008154
  44   44.20137456045023   524     0.2013745604502300
  45   44.20137456045023   524    -0.7986254395497700
  46   46.10105039582103   598     0.1010503958210259
  47   46.10105039582103   598    -0.8989496041789741
  48   48.26520579699401   692     0.2652057969940090
  49   48.26520579699401   692    -0.7347942030059910
  50   50.20979388257698   786     0.2097938825769775
  51   50.20979388257698   786    -0.7902061174230225
  52   52.33643071077409   900     0.3364307107740934
  53   52.33643071077409   900    -0.6635692892259066
  54   54.26025209047609  1014     0.2602520904760932
  55   54.26025209047609  1014    -0.7397479095239068
  56   56.40159041927338  1154     0.4015904192733788
  57   56.40159041927338  1154    -0.5984095807266212
  58   58.34609979287966  1294     0.3460997928796596
  59   58.34609979287966  1294    -0.6539002071203404
  60   60.44611088173228  1460     0.4461108817322844
  61   60.44611088173228  1460    -0.5538891182677156
  62   62.36400319817613  1626     0.3640031981761257
  63   62.36400319817613  1626    -0.6359968018238743
  64   64.49766201469299  1828     0.4976620146929893
  65   64.49766201469299  1828    -0.5023379853070107
  66   66.45061225735103  2030     0.4506122573510340
  67   66.45061225735103  2030    -0.5493877426489660
  68   68.56130277140938  2268     0.5613027714093829
  69   68.56130277140938  2268    -0.4386972285906171


In reverse it looks like
Code:
.        

     |  a^ =               |      |                      
  k  |  sinh°05(k/2+1/2)   | A[k] |   r = a^ /A[k] - 1
-----+---------------------+------+-------------------
   0   0.5102310310942976    1    -0.4897689689057024
   1   1.078138996064359     1     0.0781389960643593
   2   1.748304390930525     2    -0.1258478045347374
   3   2.552519461332870     2     0.2762597306664349
   4   3.515579526352944     4    -0.1211051184117640
   5   4.659355506757197     4     0.1648388766892992
   6   6.004813731822326     6  0.0008022886370542993
   7   7.572966044912433     6     0.2621610074854055
   8   9.385336346538606    10   -0.06146636534613942
   9   11.46420789661713    10     0.1464207896617127
  10   13.83276703665167    14   -0.01194521166773818
  11   16.51519557483509    14     0.1796568267739347
  12   19.53673656870888    20   -0.02316317156455596
  13   22.92374579523910    20     0.1461872897619548
  14   26.70373529679550    26    0.02706674218444246
  15   30.90541246272600    26     0.1886697101048460
  16   35.55871659041376    36   -0.01225787248850666
  17   40.69485405483909    36     0.1304126126344191
  18   46.34633276239224    46   0.007528973095483409
  19   52.54699630468353    46     0.1423260066235550
  20   59.33205807477817    60   -0.01113236542036384
  21   66.73813551558644    60     0.1123022585931074
  22   74.80328461281173    74    0.01085519747042881
  23   83.56703470866675    74     0.1292842528198210
  24   93.07042368928640    94  -0.009889109688442508
  25   103.3560335835290    94    0.09953227216520219
  26   114.4680266007278   114   0.004105496497612463
  27   126.4521816281218   114     0.1092296634045772
  28   139.3559312040241   140  -0.004600491399828113
  29   153.2283989795549   140    0.09448856413967771
  30   168.1204376795053   166    0.01277372096087535
  31   184.0846675712996   166     0.1089437805499975
  32   201.1755154498853   202  -0.004081606683735970
  33   219.4492541455617   202    0.08638244626515684
  34   238.9640425611633   238   0.004050598996484624
  35   259.7799662445876   238    0.09151246321255298
  36   281.9590785023372   284  -0.007186343301629582
  37   305.5654420595219   284    0.07593465513916145
  38   330.6651712715941   330   0.002015670519982010
  39   357.3264748929723   330    0.08280749967567349
  40   385.6196994076169   390   -0.01123153998046941
  41   415.6173729265633   390    0.06568557160657261
  42   447.3942496573710   450  -0.005790556316953349
  43   481.0273549504231   450    0.06894967766760693
  44   516.5960309269911   524   -0.01412971197139100
  45   554.1819826939725   524    0.05759920361445136
  46   593.8693251502086   598  -0.006907483026407088
  47   635.7446303892922   598    0.06311811101888332
  48   679.8969757037854   692   -0.01748991950320035
  49   726.4179921957754   692    0.04973698294187201
  50   775.4019139987164   786   -0.01348356997618778
  51   826.9456281155144   786    0.05209367444721939
  52   881.1487248778374   900   -0.02094586124684738
  53   938.1135490316460   900    0.04234838781293998
  54   997.9452514539660  1014   -0.01583308535111831
  55   1060.751841505941  1014    0.04610635256996131
  56   1126.644240027227  1154   -0.02370516462112075
  57   1195.736332976817  1154    0.03616666635772742
  58   1268.145025725405  1294   -0.01998066018129422
  59   1343.990298004411  1294    0.03863237867419740
  60   1423.395259516844  1460   -0.02507174005695639
  61   1506.486206215162  1460    0.03183986727065867
  62   1593.392677251357  1626   -0.02005370402745561
  63   1684.247512604481  1626    0.03582257847754043
  64   1779.186911390872  1828   -0.02670300252140491
  65   1878.350490862376  1828    0.02754403220042451
  66   1981.881346097857  2030   -0.02370377039514422
  67   2089.926110393341  2030    0.02952025142529100
  68   2202.635016356147  2268   -0.02882053952550826
  69   2320.161957708406  2268    0.02299909951869768


Perhaps it is easy to finetune this much more with not too much effort.

[update]: a first finetuning: a^ (k) = sinh°05( (1.018 k -0.174)/2)
See the pictures with the already improved data.

Absolute values:
   

Ratios: [update]: upps, after looking behind index k>150 things are no more looking so nice... :-(
   


Technical details:
I computed the powerseries for the half-iterate from the sqrt of the carlemanmatrix for sinh to 128 terms.
The convergence-radius is zero, but for x near zero one can evaluate the first 128 terms of the powerseries and get a value, which reinserted gives indeed the one-time-iterate to, say 20 or 30 dec digits precision.
So high arguments x of the sinh°05(x) must be transferred by asinh()-iterations sufficiently near towards zero (say 0<x<0.5 or x<0.2 for k~80) , then the powerseries can be evaluated with that x, and then the result must be retransferred by appropriate sinh()-iterations.
I computed this with 400 digits internal precision and float algebra (although the square-root of the Carlemanmatrix can be determined in rational algebra).

Gottfried

(If someone needs it I can supply the Pari/GP-code)
Gottfried Helms, Kassel
Reply


Messages In This Thread
My favorite integer sequence - by tommy1729 - 07/27/2014, 08:44 PM
RE: My favorite integer sequence - by jaydfox - 07/31/2014, 01:51 AM
RE: My favorite integer sequence - by jaydfox - 08/01/2014, 01:53 AM
RE: My favorite integer sequence - by Gottfried - 08/01/2014, 03:25 PM
RE: My favorite integer sequence - by jaydfox - 08/01/2014, 05:04 PM
RE: My favorite integer sequence - by sheldonison - 08/01/2014, 11:36 PM
RE: My favorite integer sequence - by jaydfox - 08/01/2014, 11:57 PM
RE: My favorite integer sequence - by jaydfox - 08/05/2014, 04:49 PM
RE: My favorite integer sequence - by jaydfox - 08/05/2014, 05:57 PM
RE: My favorite integer sequence - by Gottfried - 08/06/2014, 04:38 PM
RE: My favorite integer sequence - by jaydfox - 08/07/2014, 01:19 AM
RE: My favorite integer sequence - by jaydfox - 08/07/2014, 01:34 AM
RE: My favorite integer sequence - by jaydfox - 08/09/2014, 09:42 AM
RE: My favorite integer sequence - by Gottfried - 08/09/2014, 02:27 PM
RE: My favorite integer sequence - by tommy1729 - 09/09/2014, 12:55 AM
RE: My favorite integer sequence - by jaydfox - 08/08/2014, 12:55 AM
RE: My favorite integer sequence - by Gottfried - 08/08/2014, 02:27 AM
RE: My favorite integer sequence - by jaydfox - 09/09/2014, 07:43 PM
RE: My favorite integer sequence - by jaydfox - 09/09/2014, 09:45 PM
RE: My favorite integer sequence - by jaydfox - 08/02/2014, 12:08 AM
RE: My favorite integer sequence - by tommy1729 - 08/03/2014, 11:38 PM
RE: My favorite integer sequence - by sheldonison - 08/04/2014, 11:49 PM
RE: My favorite integer sequence - by jaydfox - 09/16/2014, 05:32 AM
RE: My favorite integer sequence - by Gottfried - 09/17/2014, 07:39 PM
RE: My favorite integer sequence - by jaydfox - 10/02/2014, 10:53 PM
RE: My favorite integer sequence - by Gottfried - 08/03/2014, 03:32 PM
RE: My favorite integer sequence - by tommy1729 - 08/03/2014, 11:44 PM
RE: My favorite integer sequence - by sheldonison - 08/02/2014, 05:48 AM
RE: My favorite integer sequence - by tommy1729 - 09/10/2014, 08:57 PM
RE: My favorite integer sequence - by Gottfried - 08/02/2014, 07:43 PM
RE: My favorite integer sequence - by Gottfried - 08/02/2014, 09:29 PM
RE: My favorite integer sequence - by Gottfried - 08/02/2014, 09:36 PM
RE: My favorite integer sequence - by tommy1729 - 08/05/2014, 11:16 PM
RE: My favorite integer sequence - by tommy1729 - 08/08/2014, 11:02 PM
RE: My favorite integer sequence - by jaydfox - 08/09/2014, 07:02 PM
RE: My favorite integer sequence - by jaydfox - 08/09/2014, 10:51 PM
RE: My favorite integer sequence - by sheldonison - 08/11/2014, 04:51 PM
RE: My favorite integer sequence - by jaydfox - 08/11/2014, 05:19 PM
RE: My favorite integer sequence - by jaydfox - 08/19/2014, 01:36 AM
RE: My favorite integer sequence - by jaydfox - 08/19/2014, 02:05 AM
RE: My favorite integer sequence - by jaydfox - 08/19/2014, 05:31 PM
RE: My favorite integer sequence - by sheldonison - 08/19/2014, 07:56 PM
RE: My favorite integer sequence - by jaydfox - 08/20/2014, 07:42 AM
RE: My favorite integer sequence - by sheldonison - 08/20/2014, 02:11 PM
RE: My favorite integer sequence - by jaydfox - 08/20/2014, 07:57 PM
RE: My favorite integer sequence - by jaydfox - 08/21/2014, 01:15 AM
RE: My favorite integer sequence - by jaydfox - 08/21/2014, 05:25 AM
RE: My favorite integer sequence - by jaydfox - 08/22/2014, 05:39 PM
RE: My favorite integer sequence - by jaydfox - 09/11/2014, 01:33 AM
RE: My favorite integer sequence - by tommy1729 - 08/09/2014, 09:16 PM
RE: My favorite integer sequence - by jaydfox - 08/09/2014, 10:19 PM
RE: My favorite integer sequence - by tommy1729 - 08/09/2014, 10:52 PM
RE: My favorite integer sequence - by jaydfox - 08/09/2014, 11:46 PM
RE: My favorite integer sequence - by tommy1729 - 08/09/2014, 11:10 PM
RE: My favorite integer sequence - by jaydfox - 08/10/2014, 12:30 AM
RE: My favorite integer sequence - by tommy1729 - 08/11/2014, 12:17 PM
RE: My favorite integer sequence - by Gottfried - 08/22/2014, 12:30 AM
Amazing variant - by tommy1729 - 08/26/2014, 08:57 PM
RE: My favorite integer sequence - by tommy1729 - 09/01/2014, 10:37 PM
RE: My favorite integer sequence - by tommy1729 - 10/02/2014, 11:24 PM
RE: My favorite integer sequence - by jaydfox - 10/02/2014, 11:29 PM
RE: My favorite integer sequence - by tommy1729 - 02/10/2015, 12:15 AM
RE: My favorite integer sequence - by tommy1729 - 02/15/2015, 05:19 PM
RE: My favorite integer sequence - by tommy1729 - 10/07/2015, 08:22 AM
RE: My favorite integer sequence - by tommy1729 - 10/07/2015, 09:10 PM
RE: My favorite integer sequence - by tommy1729 - 03/13/2016, 12:31 AM

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