Series

gAr · 2012-10-20 23:19:41

Hi,

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

bobbym · 2012-10-20 23:26:49

Hi gAr;

What is this sequence related to?

phrontister · 2012-10-20 23:38:21

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.

gAr · 2012-10-20 23:40:24

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)

bobbym · 2012-10-20 23:47:03

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

gAr · 2012-10-20 23:52:13

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

bobbym · 2012-10-20 23:56:02

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?

phrontister · 2012-10-21 00:14:02

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

bobbym · 2012-10-21 00:21:04

Hi;

What pattern did you spot?

phrontister · 2012-10-21 00:26:56

Hi Bobby,

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

gAr · 2012-10-21 00:27:51

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.

bobbym · 2012-10-21 00:29:45

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!

gAr · 2012-10-21 00:51:19

Hi bobbym,

I suspect gAr got it too.

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

bobbym · 2012-10-21 01:33:52

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, ...

gAr · 2012-10-21 01:54:25

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.

bobbym · 2012-10-21 02:23:53

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.

anonimnystefy · 2012-11-24 16:56:34

gAr wrote:

Hi anonimnystefy,
Which is that problem?

I need a hint on sum 23 on page 12.

gAr · 2012-11-24 17:13:04

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.

bobbym · 2012-11-24 17:14:56

Hi gAr;

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

gAr · 2012-11-25 00:23:29

Hi bobbym,

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

bobbym · 2012-11-25 00:33:29

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!

gAr · 2012-11-25 00:55:39

That's true!

bobbym · 2012-11-25 01:02:50

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

anonimnystefy · 2012-11-25 01:04:41

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...

gAr · 2012-11-25 04:11:59

Hi bobbym,

Yes, I will be continuing with it later.

Hi anonimnystefy,

You too used g.fs?

Math Is Fun Forum

#476 2012-10-20 23:19:41

Re: Series

#477 2012-10-20 23:26:49

Re: Series

#478 2012-10-20 23:38:21

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#479 2012-10-20 23:40:24

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#480 2012-10-20 23:47:03

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#481 2012-10-20 23:52:13

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#482 2012-10-20 23:56:02

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#483 2012-10-21 00:14:02

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#484 2012-10-21 00:21:04

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#485 2012-10-21 00:26:56

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#486 2012-10-21 00:27:51

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#487 2012-10-21 00:29:45

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#488 2012-10-21 00:51:19

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#489 2012-10-21 01:33:52

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#490 2012-10-21 01:54:25

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#491 2012-10-21 02:23:53

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#492 2012-11-24 16:56:34

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#493 2012-11-24 17:13:04

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#494 2012-11-24 17:14:56

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#495 2012-11-25 00:23:29

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#496 2012-11-25 00:33:29

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#497 2012-11-25 00:55:39

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#498 2012-11-25 01:02:50

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#499 2012-11-25 01:04:41

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#500 2012-11-25 04:11:59

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