[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,
>
>
>
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 ...