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#476 2012-10-21 22:19:41

gAr
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Re: Series

Hi,

Ah, got it. Started my index with a zero, so it went wrong.
Very good answer!


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."
 

#477 2012-10-21 22:26:49

bobbym
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Re: Series

Hi gAr;

What is this sequence related to?


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
 

#478 2012-10-21 22:38:21

phrontister
Real Member

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Re: Series

Hi Bobby & gAr,

I found a recurrence with Excel and used that to find the first 120 numbers.



But I couldn't work out how to express that in a formula.


"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson
 

#479 2012-10-21 22:40:24

gAr
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Re: Series

Hi bobbym,

This is from proj. eul.
I was stuck with that series, but it does not end there yet, I need to generalize it to a higher degree. ( I mean there are more series )
Newton's difference formulas did not work.
How did you get the recurrence?

Hi phrontister,

Good to see you!
Yes, those terms are correct, good try!

Last edited by gAr (2012-10-21 22:45:09)


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."
 

#480 2012-10-21 22:47:03

bobbym
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Re: Series

Mathematica has tremendous capabilities to find recurrences and sequence functions. I just had to coax it some and then clean up the mess.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
 

#481 2012-10-21 22:52:13

gAr
Star Member

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Re: Series

It can get you the recurrence from a series? That's a very good feature indeed!


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."
 

#482 2012-10-21 22:56:02

bobbym
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Re: Series

I am not sure it can. I just used your sequence. It uses some routines from DZ to get a difference equation. Isn't this problem a little tough for project Euler?


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
 

#483 2012-10-21 23:14:02

phrontister
Real Member

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Re: Series

Here's my spreadsheet result, with the sequence in column A.

Columns B, C & D contain the differences between the numbers of the preceding column in a particular row and the previous one.

Once I'd spotted the pattern I ploughed it back into the spreadsheet from the 10th row (with 133 in column A) onwards.


Code:

     A       B     C     D
     1            
     1       0        
     4       3     3    
     8       4     1    -2
    16       8     4     3
    28      12     4     0
    45      17     5     1
    67      22     5     0
    97      30     8     3
   133      36     6    -2
   178      45     9     3
   232      54     9     0
   296      64    10     1
   370      74    10     0
   457      87    13     3
   555      98    11    -2
   667     112    14     3
   793     126    14     0
   934     141    15     1
  1090     156    15     0
  1264     174    18     3
  1454     190    16    -2
  1663     209    19     3
  1891     228    19     0
  2139     248    20     1
  2407     268    20     0
  2698     291    23     3
  3010     312    21    -2
  3346     336    24     3
  3706     360    24     0
  4091     385    25     1
  4501     410    25     0
  4939     438    28     3
  5403     464    26    -2
  5896     493    29     3
  6418     522    29     0
  6970     552    30     1
  7552     582    30     0
  8167     615    33     3
  8813     646    31    -2
  9493     680    34     3
 10207     714    34     0
 10956     749    35     1
 11740     784    35     0
 12562     822    38     3
 13420     858    36    -2
 14317     897    39     3
 15253     936    39     0
 16229     976    40     1
 17245    1016    40     0
 18304    1059    43     3
 19404    1100    41    -2
 20548    1144    44     3
 21736    1188    44     0
 22969    1233    45     1
 24247    1278    45     0
 25573    1326    48     3
 26945    1372    46    -2
 28366    1421    49     3
 29836    1470    49     0
 31356    1520    50     1
 32926    1570    50     0
 34549    1623    53     3
 36223    1674    51    -2
 37951    1728    54     3
 39733    1782    54     0
 41570    1837    55     1
 43462    1892    55     0
 45412    1950    58     3
 47418    2006    56    -2
 49483    2065    59     3
 51607    2124    59     0
 53791    2184    60     1
 56035    2244    60     0
 58342    2307    63     3
 60710    2368    61    -2
 63142    2432    64     3
 65638    2496    64     0
 68199    2561    65     1
 70825    2626    65     0
 73519    2694    68     3
 76279    2760    66    -2
 79108    2829    69     3
 82006    2898    69     0
 84974    2968    70     1
 88012    3038    70     0
 91123    3111    73     3
 94305    3182    71    -2
 97561    3256    74     3
100891    3330    74     0
104296    3405    75     1
107776    3480    75     0
111334    3558    78     3
114968    3634    76    -2
118681    3713    79     3
122473    3792    79     0
126345    3872    80     1
130297    3952    80     0
134332    4035    83     3
138448    4116    81    -2
142648    4200    84     3
146932    4284    84     0
151301    4369    85     1
155755    4454    85     0
160297    4542    88     3
164925    4628    86    -2
169642    4717    89     3
174448    4806    89     0
179344    4896    90     1
184330    4986    90     0
189409    5079    93     3
194579    5170    91    -2
199843    5264    94     3
205201    5358    94     0
210654    5453    95     1
216202    5548    95     0
221848    5646    98     3
227590    5742    96    -2
233431    5841    99     3
239371    5940    99     0

"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson
 

#484 2012-10-21 23:21:04

bobbym
Administrator

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Re: Series

Hi;

What pattern did you spot?


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
 

#485 2012-10-21 23:26:56

phrontister
Real Member

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Re: Series

Hi Bobby,

The pattern in column D that recurs every sixth row: -2, 3, 0, 1, 0, 3


"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson
 

#486 2012-10-21 23:27:51

gAr
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Re: Series

Hi phrontister,

A table like that is exactly what is used for finding polynomial's formula.

Hi bobbym,

He's talking about col. D.

Yeah, there are many more like that. There are few problems on expectation and probability there, interesting.


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."
 

#487 2012-10-21 23:29:45

bobbym
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Re: Series

Hi;

Yes, I see your idea now and it gives me an idea.

A brilliant stroke by phrontister. I suspect gAr got it too. If I only would went one deeper I would have also. Anyways, it makes the solution very much simpler!


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
 

#488 2012-10-21 23:51:19

gAr
Star Member

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Re: Series

Hi bobbym,

I suspect gAr got it too.

Yes, I was referring to that column in #468.


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."
 

#489 2012-10-22 00:33:52

bobbym
Administrator

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Re: Series

The new solution;

If you break up the sequence into parts you get a much simpler answer. For instance:
y1, y7, y13...





y2, y8, y14...





y3, y9, y15...





y4, y10, y16...





y5, y11, y17...





y6, y12, y18, ...




In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
 

#490 2012-10-22 00:54:25

gAr
Star Member

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Re: Series

Hi bobbym,

That's a good idea!
Any trick to combine them?
Because I think the length of the cycle will only get larger, e.g. the next series I have is
1, 1, 1, 5, 14, 29, 55, 97, 157, 241, 355, 505, 698, 941, 1241, 1609, 2053, 2581, 3205, 3937, 4786, 5765, 6887, 8165, 9613, 11245, 13075, 15121, 17398, 19921, 22709, 25781, 29153, 32845, 36877,
41269, 46042, 51217, 56815, 62861, 69377, 76385, 83911, 91981, 100618, 109849, 119701, 130201, 141377, 153257, 165869, 179245, 193414, 208405, 224251, 240985, 258637, 277241, 296831, 317441, 339106, 361861,...


Anyway, I'll look into it when I get some free time. See you later.


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."
 

#491 2012-10-22 01:23:53

bobbym
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Re: Series

Hi gAr;

The purpose of splitting them is to make them simpler. If we combine them they should get ugly like the one I gave in an earlier post. But I will see if I can get something simpler than that answer.

For your new sequence:



This sequence has a period of 12. Making the approach I used in the last one too cumbersome.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
 

#492 2012-11-25 15:56:34

anonimnystefy
Real Member

Online

Re: Series

gAr wrote:

Hi anonimnystefy,

Which is that problem?

I need a hint on sum 23 on page 12.


The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
 

#493 2012-11-25 16:13:04

gAr
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Re: Series

Hi anonimnystefy,

I think I got that when trying out some convolution of g.fs. I'm not sure where I wrote them, I too may need to work on the problem!


Hi bobbym,

Thanks for the formula.


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."
 

#494 2012-11-25 16:14:56

bobbym
Administrator

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Re: Series

Hi gAr;

It is not nice but I could not come up with something cleaner.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
 

#495 2012-11-25 23:23:29

gAr
Star Member

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Re: Series

Hi bobbym,

It's okay, at least it has shown me that a formula exists for such kind.


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."
 

#496 2012-11-25 23:33:29

bobbym
Administrator

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Re: Series

Hi;

There must be formulas for lots of things we do not know. They may be hundreds of terms long and
therefore impossible to discover. We need faster computers!


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
 

#497 2012-11-25 23:55:39

gAr
Star Member

Offline

Re: Series

That's true!


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."
 

#498 2012-11-26 00:02:50

bobbym
Administrator

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Re: Series

Are you still working on the problem that you needed the formula for that sequence?


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
 

#499 2012-11-26 00:04:41

anonimnystefy
Real Member

Online

Re: Series

gAr wrote:

Hi anonimnystefy,

I think I got that when trying out some convolution of g.fs. I'm not sure where I wrote them, I too may need to work on the problem!


Hi bobbym,

Thanks for the formula.

Hi gAr

Thank you!! I finally managed to do it! It seems very hard to do by any other method...


The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
 

#500 2012-11-26 03:11:59

gAr
Star Member

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Re: Series

Hi bobbym,

Yes, I will be continuing with it later.

Hi anonimnystefy,

You too used g.fs?


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."
 

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