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**gAr****Member**- Registered: 2011-01-09
- Posts: 3,479

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."

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,771

Hi gAr;

What is this sequence related to?

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**phrontister****Real Member**- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 3,848

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

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**gAr****Member**- Registered: 2011-01-09
- Posts: 3,479

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 (2012-10-20 23:45:09)*

"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."

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,771

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.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**gAr****Member**- Registered: 2011-01-09
- Posts: 3,479

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."

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,771

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.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**phrontister****Real Member**- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 3,848

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.

```
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

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,771

Hi;

What pattern did you spot?

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**phrontister****Real Member**- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 3,848

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

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**gAr****Member**- Registered: 2011-01-09
- Posts: 3,479

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.

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

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,771

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!

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**gAr****Member**- Registered: 2011-01-09
- Posts: 3,479

Hi bobbym,

I suspect gAr got it too.

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

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

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,771

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, ...

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**gAr****Member**- Registered: 2011-01-09
- Posts: 3,479

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.

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

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,771

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.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,526

gAr wrote:

Hi anonimnystefy,

Which is that problem?

I need a hint on sum 23 on page 12.

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**gAr****Member**- Registered: 2011-01-09
- Posts: 3,479

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.

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

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,771

Hi gAr;

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

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**gAr****Member**- Registered: 2011-01-09
- Posts: 3,479

Hi bobbym,

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

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

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,771

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!

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**gAr****Member**- Registered: 2011-01-09
- Posts: 3,479

That's true!

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

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,771

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

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,526

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...

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**gAr****Member**- Registered: 2011-01-09
- Posts: 3,479

Hi bobbym,

Yes, I will be continuing with it later.

Hi anonimnystefy,

You too used g.fs?

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

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