[seqfan] Re: Fw: Closed form?

Prof. Dr. Alois Heinz heinz at hs-heilbronn.de
Thu Apr 30 20:59:57 CEST 2009


David Wilson schrieb:

>At n = 10^30, n is around 1359 digits (according to your calculations). 
>For such large n, I would expect the jaggies in a(n) to have smoothed 
>out, and at least the initial, say, 100 digits of a(n) to be growing in 
>a steady fashion, similarly for f(n). The jaggies in a(n)/f(n) in the 
>5th digit (it maxes at n = 10^27) therefore seem suspicious to me. I 
>would expect a(n)/f(n) to drift in a constant direction, either up or 
>down, in [10^25,10^30] range. I suspect roundoff errors, etc, but still, 
>the fact that we get .9233 over this huge range almost vindicates my 
>suspicions.
>  
>
Your observation is correct.  It is perhaps better here to do the
calculation with powers of 2 instead of 10, because the sequence a is

   A000123 Number of partitions of 2n into powers of 2.  

I have looked at a larger range now, and a(n)/f(n) is drifting slowly
upwards or downwards, always with prefix .9233

Digits:=160:
...
for p from 80 to 200 do m:= 2^p: print (p,"==> ",   evalf ( a(m)/f(m), 
10)  ) od:

   80, "==> ", .9233047540
   81, "==> ", .9233050280
   82, "==> ", .9233053271
   83, "==> ", .9233056462
   84, "==> ", .9233059804
   85, "==> ", .9233063246
   86, "==> ", .9233066738
   87, "==> ", .9233070239
   88, "==> ", .9233073703
   89, "==> ", .9233077100
   90, "==> ", .9233080389
   91, "==> ", .9233083536
   92, "==> ", .9233086519
   93, "==> ", .9233089313
   94, "==> ", .9233091893
   95, "==> ", .9233094244
   96, "==> ", .9233096358
   97, "==> ", .9233098214
   98, "==> ", .9233099812
   99, "==> ", .9233101148
  100, "==> ", .9233102217
  101, "==> ", .9233103018
  102, "==> ", .9233103557
  103, "==> ", .9233103842
  104, "==> ", .9233103876
  105, "==> ", .9233103668
  106, "==> ", .9233103234
  107, "==> ", .9233102580
  108, "==> ", .9233101724
  109, "==> ", .9233100669
  110, "==> ", .9233099439
  111, "==> ", .9233098053
  112, "==> ", .9233096517
  113, "==> ", .9233094845
  114, "==> ", .9233093060
  115, "==> ", .9233091184
  116, "==> ", .9233089220
  117, "==> ", .9233087189
  118, "==> ", .9233085108
  119, "==> ", .9233082983
  120, "==> ", .9233080847
  121, "==> ", .9233078692
  122, "==> ", .9233076545
  123, "==> ", .9233074413
  124, "==> ", .9233072316
  125, "==> ", .9233070254
  126, "==> ", .9233068243
  127, "==> ", .9233066297
  128, "==> ", .9233064411
  129, "==> ", .9233062608
  130, "==> ", .9233060885
  131, "==> ", .9233059260
  132, "==> ", .9233057729
  133, "==> ", .9233056299
  134, "==> ", .9233054978
  135, "==> ", .9233053763
  136, "==> ", .9233052664
  137, "==> ", .9233051679
  138, "==> ", .9233050813
  139, "==> ", .9233050063
  140, "==> ", .9233049423
  141, "==> ", .9233048908
  142, "==> ", .9233048504
  143, "==> ", .9233048225
  144, "==> ", .9233048058
  145, "==> ", .9233047996
  146, "==> ", .9233048045
  147, "==> ", .9233048206
  148, "==> ", .9233048463
  149, "==> ", .9233048824
  150, "==> ", .9233049278
  151, "==> ", .9233049824
  152, "==> ", .9233050460
  153, "==> ", .9233051177
  154, "==> ", .9233051970
  155, "==> ", .9233052838
  156, "==> ", .9233053773
  157, "==> ", .9233054773
  158, "==> ", .9233055834
  159, "==> ", .9233056944
  160, "==> ", .9233058103
  161, "==> ", .9233059309
  162, "==> ", .9233060550
  163, "==> ", .9233061825
  164, "==> ", .9233063130
  165, "==> ", .9233064455
  166, "==> ", .9233065804
  167, "==> ", .9233067164
  168, "==> ", .9233068534
  169, "==> ", .9233069911
  170, "==> ", .9233071285
  171, "==> ", .9233072661
  172, "==> ", .9233074027
  173, "==> ", .9233075379
  174, "==> ", .9233076720
  175, "==> ", .9233078040
  176, "==> ", .9233079338
  177, "==> ", .9233080613
  178, "==> ", .9233081857
  179, "==> ", .9233083071
  180, "==> ", .9233084253
  181, "==> ", .9233085395
  182, "==> ", .9233086496
  183, "==> ", .9233087565
  184, "==> ", .9233088583
  185, "==> ", .9233089563
  186, "==> ", .9233090496
  187, "==> ", .9233091384
  188, "==> ", .9233092213
  189, "==> ", .9233092999
  190, "==> ", .9233093729
  191, "==> ", .9233094411
  192, "==> ", .9233095038
  193, "==> ", .9233095612
  194, "==> ", .9233096133
  195, "==> ", .9233096600
  196, "==> ", .9233097009
  197, "==> ", .9233097367
  198, "==> ", .9233097677
  199, "==> ", .9233097929
  200, "==> ", .9233098129

# range of decimal digits in a(n):

length (a(2^80));     862
length (a(2^200));    5684

>Thank you for your interest. If you make any discoveries of interest, 
>please feel free to publish, I won't ask for attribution.
>
>  
>
Thank you.  But up to now we only have interesting observations. 
A closed formula or a proof that 0.9233... is the correct factor would 
by nice ...








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