Difference between difference down to 1.

[sunshaker]

I have been playing about with primes, But this is not a prime question, I was looking at the difference between each prime, Then the difference between that difference until I reached 1, To look for patterns.

 

I was wondering what it is called when you take a certain amount of numbers has shown below and find the difference between the difference until you end with 1? (1 is the point of triangle).

 

I have tried to look it up but end up with differences between everything but what I want to know, There is most probably a simple name but it escapes me at the moment.

 

 

Each difference of the same number is the same colour,0 pink,1 purple, 2 blue, 4 yellow.

[mathematic]

I can't get your picture to display.

[sunshaker]

I can't get your picture to display.

Hope this is better,

Numbers are hard to see, Better in " open in new tab".

 

example, difference between each number until 1 point of triangle.

 

2

1

3 1

2 1

5 0 1

2 2

7 2

4

11

[Acme]

I would call them 1-dimensional cellular automata. The difference part is simply the 'rule' and the Primes the 'initial condition(s)'.

Addendum: Try using some other sets for your initial condition, such as Perfect Squares, Fibonacci numbers, Triangular numbers, etcetera.

[sunshaker]

I would call them 1-dimensional cellular automata. The difference part is simply the 'rule' and the Primes the 'initial condition(s)'.

Addendum: Try using some other sets for your initial condition, such as Perfect Squares, Fibonacci numbers, Triangular numbers, etcetera.

Thanks Acme, At least now I have a name to call them, "Dimensional cellular automata art".

 

Just finished extending the primes, I was thinking what set of numbers to use next, Thanks for ideas, I think next I will do Fibonacci numbers. I find it relaxing and time flies .

 

larger set of primes

Triangular numbers soon came to an end

 

[imatfaal]

It is much more fun to work out what will happen in your head. Squares are also very boring - finish just as quickly but with a column of twos

[sunshaker]

It is much more fun to work out what will happen in your head. Squares are also very boring - finish just as quickly but with a column of twos

As i have just found out

 

Any ideas on a good set of numbers?

 

Fibonacci numbers are alright, Each difference line are Fib numbers moved down a place, Until it becomes ones and zeros " The hidden programmers binary code of our universe"

 

 

Composite Numbers

 

Primes are still the most pleasing up to yet.

Edited July 3, 2014 by sunshaker

[Acme]

Thanks Acme, At least now I have a name to call them, "Dimensional cellular automata art".

 

Just finished extending the primes, I was thinking what set of numbers to use next, Thanks for ideas, I think next I will do Fibonacci numbers. I find it relaxing and time flies .

Just to clarify, they are "1-dimensional" cellular automata. This is because the initial-condition set is on a 1-dimensional line. Conway's Game of Life on the other hand is a 2-dimensional cellular automaton because the initial condition set is on a table of rows & columns.

 

Anyway, try this set: {77 119 143 161 209 221 299 323 329 371 377 407 413 437 473 497 527 533 539 551 589 611 623 629 689 707 ...} These are the ordered odd numbers that are not polygonal numbers and not Prime.

 

Or this set: {8 14 20 26 32 38 44 50 56 62 68 74 80 86 92 98 104 110 116 122 128 ...} These are the ordered even numbers that are not polygonal numbers. (All are of form 6x+2)

 

Good times.

.

Addendum: Try the Lucas Numbers too. {2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843 ...}

[sunshaker]

 

Just to clarify, they are "1-dimensional" cellular automata. This is because the initial-condition set is on a 1-dimensional line. Conway's Game of Life on the other hand is a 2-dimensional cellular automaton because the initial condition set is on a table of rows & columns.

 

Anyway, try this set: {77 119 143 161 209 221 299 323 329 371 377 407 413 437 473 497 527 533 539 551 589 611 623 629 689 707 ...} These are the ordered odd numbers that are not polygonal numbers and not Prime.

 

Or this set: {8 14 20 26 32 38 44 50 56 62 68 74 80 86 92 98 104 110 116 122 128 ...} These are the ordered even numbers that are not polygonal numbers. (All are of form 6x+2)

 

Good times.

I tried the first set, I thought they would break down quicker into 0,1,2,3s, It seems to me at the moment that the "primes" bring their own beauty to a sequence/set of numbers.

 

The second set will break down into 6's then 0's within 2 differences.

 

Anymore interesting sets of numbers appreciated

 

Edit a few more colours

 

Edited July 3, 2014 by sunshaker

[Acme]

I tried the first set, I thought they would break down quicker into 0,1,2,3s, It seems to me at the moment that the "primes" bring their own beauty to a sequence/set of numbers.

acme 1.png

 

The second set will break down into 6's then 0's within 2 differences.

 

Anymore interesting sets of numbers appreciated

Cool beans. You may have missed the Lucas Numbers appended to my earlier post. This appending automatically joins 2 recent posts into 1, and so such posts are often overlooked.

Those who can't count, don't count.

 

PS Maybe add a few more colors for 6,7,8...?

Edited July 3, 2014 by Acme

[sunshaker]

Cool beans. You may have missed the Lucas Numbers appended to my earlier post. This appending automatically joins 2 recent posts into 1, and so such posts are often overlooked.

Those who can't count, don't count.

 

PS Maybe add a few more colors for 6,7,8...?

I have added a few more colours,

 

I have learnt more about numbers in the last couple of weeks than I have done all my life,

This is similar to how I learnt the periodic table, You learn without realizing you are learning, As soon as I started doing the Lucas numbers I realized how similar to the Fibonacci sequence they are, Which gets you thinking on other ways to create sequences.

 

Now I am hungry for sequences to explore, Or Mixing sequences.

 

Lucas

[Acme]

...

Now I am hungry for sequences to explore, Or Mixing sequences.

...

Feeeed me!

 

How about the sequence of Prime Gaps? These are distances/differences between Primes. {1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, ...}

 

Longer list here: >> Prime Gaps @ OEIS

.

Addendum: Speaking of ending with 1, you may have some fun with the hailstone numbers.

Collatz Conjecture

The Collatz conjecture is a conjecture in mathematics named after Lothar Collatz, who first proposed it in 1937. The conjecture is also known as the 3n + 1 conjecture, the Ulam conjecture (after Stanisław Ulam), Kakutani's problem (after Shizuo Kakutani), the Thwaites conjecture (after Sir Bryan Thwaites), Hasse's algorithm (after Helmut Hasse), or the Syracuse problem;[1][2] the sequence of numbers involved is referred to as the hailstone sequence or hailstone numbers (because the values are usually subject to multiple descents and ascents like hailstones in a cloud),[3][4] or as wondrous numbers.[5]

 

Take any natural number n. If n is even, divide it by 2 to get n / 2. If n is odd, multiply it by 3 and add 1 to obtain 3n + 1. Repeat the process (which has been called "Half Or Triple Plus One", or HOTPO[6]) indefinitely. The conjecture is that no matter what number you start with, you will always eventually reach 1. The property has also been called oneness.[7]

 

Paul Erdős said about the Collatz conjecture: "Mathematics may not be ready for such problems."[8] He also offered $500 for its solution.[9]

 

In 1972, J. H. Conway proved that a natural generalization of the Collatz problem is algorithmically undecidable.[10] .

The following set are the numbers that set records for number of iterations before reaching 1. {1, 2, 3, 6, 7, 9, 18, 25, 27, 54, 73, 97, 129, 171, 231, 313, 327, 649, 703, 871, 1161, 2223, 2463, 2919, 3711, 6171, 10971, ...} source @OEIS

[sunshaker]

Feeeed me!

 

How about the sequence of Prime Gaps? These are distances/differences between Primes. {1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, ...}

 

Longer list here: >> Prime Gaps @ OEIS

.

Addendum: Speaking of ending with 1, you may have some fun with the hailstone numbers.

Collatz Conjecture

The following set are the numbers that set records for number of iterations before reaching 1. {1, 2, 3, 6, 7, 9, 18, 25, 27, 54, 73, 97, 129, 171, 231, 313, 327, 649, 703, 871, 1161, 2223, 2463, 2919, 3711, 6171, 10971, ...} source @OEIS

The "Prime Gaps" are already in with the primes, They are the second column in the "prime cellular automata"

 

I have just had ago at the "Hailstone numbers", For some reason I yet am unsure of, It ends with 2, It is the only one I have done yet that ends in two at the point, Unless I have made a mistake which I cannot yet see, But it will make me try and understand the hailstone sequence better.

 

[Acme]

The "Prime Gaps" are already in with the primes, They are the second column in the "prime cellular automata"

D'oh!

 

I have just had ago at the "Hailstone numbers", For some reason I yet am unsure of, It ends with 2, It is the only one I have done yet that ends in two at the point, Unless I have made a mistake which I cannot yet see, But it will make me try and understand the hailstone sequence better.

Trey kewl. How are you making these graphs? Would your error -if there is one- be just in entering the initial line?

 

Here's some more sequences for you.

 

Padovan sequence: {1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151, 200, 265, ...}

@ OEIS

 

Beatty sequences including:

lower Wythoff sequence @ OEIS {1, 3, 4, 6, 8, 9, 11, 12, 14, 16, 17, 19, 21, 22, 24, 25, 27, 29, ...}

upper Wythoff sequence @OEIS {2, 5, 7, 10, 13, 15, 18, 20, 23, 26, 28, 31, 34, 36, 39, 41, 44, 47, ...}

[imatfaal]

I get this impression this thread has become two addicts both feeding the other' s addiction. A perfect symbiotic relationship

[Acme]

I get this impression this thread has become two addicts both feeding the other' s addiction. A perfect symbiotic relationship

I get the impression this thread has become peepers watching two addicts feeding the other's addiction in a perfect symbiotic relationship. I need a beer to wash some of this down.

But first, this sequence is from my own research. I thought I had seen it at OEIS once, but searching just now they say no. Curiously, Wolfram on their Polygonal number page explicitly claim such numbers don't exist.

 

Sequence of numbers Polygonal in 4 ways: {225, 231, 276, 325, 435, 540, 595, 616, 651, 820, ...}

details:

 

read 225 is polygonal 4 ways; it is the 3^rd 76-sided number, the 5^th 24-sided number, the 9^th 8-sided number, and the 15^th 4 sided number.

225 4 [3, 76: 5, 24: 9, 8: 15, 4]

231 4 [3, 78: 6, 17: 11, 6: 21, 3]

276 4 [3, 93: 6, 20: 12, 6: 23, 3]

325 4 [5, 34: 10, 9: 13, 6: 25, 3]

435 4 [3, 146: 5, 45: 15, 6: 29, 3]

441 4 [3, 148: 6, 31: 9, 14: 21, 4]

540 4 [3, 181: 8, 21: 12, 10: 15, 7]

595 4 [5, 61: 7, 30: 10, 15: 34, 3]

616 4 [4, 104: 7, 31: 11, 13: 16, 7]

651 4 [3, 218: 6, 45: 14, 9: 21, 5]

820 4 [4, 138: 8, 31: 10, 20: 40, 3]

...

 

[imatfaal]

Call a number -highly polygonal if it is -polygonal in or more ways out of , 4, ... up to some limit. Then the first few 2-highly polygonal numbers up to are 1, 6, 9, 10, 12, 15, 16, 21, 28, (Sloane's A090428). Similarly, the first few 3-highly polygonal numbers up to are 1, 15, 36, 45, 325, 561, 1225, 1540, 3025, ... (Sloane's A062712). There are no 4-highly polygonal numbers of this type less than except for 1.

 

Not sure why they insist on a limit on n - but as they do your list would not comply.

[Acme]

.

Addendum: Continuing with beer in hand, Wolfram is mightily screwed up me thinks. They say:

Call a number k-highly polygonal if it is n-polygonal in k or more ways out of n=3, 4, ... up to some limit. Then the first few 2-highly polygonal numbers up to n=16 are 1, 6, 9, 10, 12, 15, 16, 21, 28, (Sloane's A090428).

Similarly, the first few 3-highly polygonal numbers up to n=16 are 1, 15, 36, 45, 325, 561, 1225, 1540, 3025, ... (Sloane's A062712).

There are no 4-highly polygonal numbers of this type less than 10^(12) except for 1.

15 is only polygonal 2 ways. 15 2 [3, 6: 5, 3] (the 3^rd 6-sided number and the 5^th 3-sided number.) 16 is polygonal just once; a square, i.e. 4-sided number.

 

Also they screw the 3-way, which may entertain & per se amuse you peepers no end. Here's my ménage à trois listing.

Sequence of numbers polygonal à trois: {36, 45, 66, 81, 105, 120, 153, 171, 190, 196, 210, 261, ...}

details:

36 3 [3, 13, 6, 4, 8, 3]

45 3 [3, 16, 5, 6, 9, 3]

66 3 [3, 23, 6, 6, 11, 3]

81 3 [3, 28, 6, 7, 9, 4]

105 3 [3, 36, 5, 12, 14, 3]

120 3 [3, 41, 8, 6, 15, 3]

153 3 [3, 52, 9, 6, 17, 3]

171 3 [3, 58, 6, 13, 18, 3]

190 3 [4, 33, 10, 6, 19, 3]

196 3 [4, 34, 7, 11, 14, 4]

210 3 [3, 71, 12, 5, 20, 3]

 

Not sure why they insist on a limit on n - but as they do your list would not comply.

It says up to some limit, and the example simply stops at 4 followed by an ellipsis.

.Addendum addendum: (curse you appending editor! )

 

Anyway, whether or not Wolfram is talking about something else than I is irrelevant to my sequences. Mine are correct as given. If Wolfram is talking about something else then by all means show how they generate the sequences and submit those sequences for Sunshaker's automaton.

[sunshaker]

.

 

 

Anyway, whether or not Wolfram is talking about something else than I is irrelevant to my sequences. Mine are correct as given. If Wolfram is talking about something else then by all means show how they generate the sequences and submit those sequences for Sunshaker's automaton.

 

I am the Automator, Resistance is futile, All shall be automated ,

 

I have already automated the valence shells of electrons,

Thinking of Ascot winners in different meetings, Also family ages, Wars, Share prices,

 

Nothing can escape "Dimensional Cellular Automata".

[Acme]

I am the Automator, Resistance is futile, All shall be automated ,

UPPER WYTHOFF.png

LOWER WYTHOFF.png

 

I have already automated the valence shells of electrons,

Thinking of Ascot winners in different meetings, Also family ages, Wars, Share prices,

 

Nothing can escape "(1)Dimensional Cellular Automata".

You da man! I like those last two a lot. I'm making up my sequence of numbers polygonal 5-ways, but I'm on beer #2 so give me a few minutes.

.

In for a penny, in for a pound. Automaton me!

 

Sequence of numbers polygonal 1 way: {1, 9, 10, 12, 16, 18, 22, 24, 25, 27, 30, 33, 34, 35, 39, 40, 42, ...}

Details:

 

6 1 [3, 3]

9 1 [3, 4]

10 1 [4, 3]

12 1 [3, 5]

16 1 [4, 4]

18 1 [3, 7]

22 1 [4, 5]

24 1 [3, 9]

25 1 [5, 4]

27 1 [3, 10]

30 1 [3, 11]

33 1 [3, 12]

34 1 [4, 7]

35 1 [5, 5]

39 1 [3, 14]

40 1 [4, 8]

42 1 [3, 15]

 

 

Sequence of numbers polygonal 2 ways: {15, 21, 28, 51, 55, 64, 70, 75, 78, 91, 100, 111, 112, 117, 126, ...}

Details:

 

15 2 [3, 6, 5, 3]

21 2 [3, 8, 6, 3]

28 2 [4, 6, 7, 3]

51 2 [3, 18, 6, 5]

55 2 [5, 7, 10, 3]

64 2 [4, 12, 8, 4]

70 2 [4, 13, 7, 5]

75 2 [3, 26, 5, 9]

78 2 [3, 27, 12, 3]

91 2 [7, 6, 13, 3]

100 2 [4, 18, 10, 4]

111 2 [3, 38, 6, 9]

112 2 [4, 20, 7, 7]

117 2 [3, 40, 9, 5]

126 2 [3, 43, 6, 10]

 

 

Sequence of numbers polygonal 5 ways:{561, 1485, 1701, 2016, 2556, 2601, 2850, 3025, 3060, 3256, 3321, 4186, 4761, 4851, 5226, ...}

Details:

 

561 5 [3, 188, 6, 39, 11, 12, 17, 6, 33, 3]

1485 5 [3, 496, 5, 150, 9, 43, 15, 16, 54, 3]

1701 5 [3, 568, 6, 115, 9, 49, 18, 13, 21, 10]

2016 5 [3, 673, 6, 136, 14, 24, 32, 6, 63, 3]

2556 5 [3, 853, 6, 172, 8, 93, 36, 6, 71, 3]

2601 5 [3, 868, 6, 175, 9, 74, 17, 21, 51, 4]

2850 5 [3, 951, 12, 45, 15, 29, 38, 6, 75, 3]

3025 5 [5, 304, 10, 69, 22, 15, 25, 12, 55, 4]

3060 5 [3, 1021, 8, 111, 15, 31, 20, 18, 24, 13]

3256 5 [4, 544, 8, 118, 11, 61, 16, 29, 22, 16]

3321 5 [3, 1108, 6, 223, 9, 94, 41, 6, 81, 3]

4186 5 [4, 699, 7, 201, 28, 13, 46, 6, 91, 3]

4761 5 [3, 1588, 6, 319, 9, 134, 18, 33, 69, 4]

4851 5 [3, 1618, 6, 325, 11, 90, 21, 25, 98, 3]

5226 5 [3, 1743, 6, 350, 12, 81, 26, 18, 39, 9]

 

[sunshaker]

I am glad I done the Polygonals, It as made me realize that it takes a certain amount of numbers in a sequence to reach 1 or 0.

 

example to end in 1 lower wythoff, must have either 3,5,6,9,15,21, numbers in a sequence to end with 1.

 

upper wythoff, must have either 3,5,6,9,19,20, " "

 

primes will end in 1 up to the first 35 primes, Then "0" at 36,38,40,44,46,48,52,54 which I must look into further.

 

lucas must have 2,3,5,6,8,9,10,11,12 to end with 1.

Which will now give me more sequences to explore.

 

The polygonals

 

All of these 1dimensional cellular automatas are in this xl.

https://alphaomegadotme.files.wordpress.com/2012/05/1-dimensional-cellular-automata-a1.xls

 

 

PS, After doing these yesterday I had a amazing dream where I was in polygonal world, Made up of all these dimensional cellular automatas, all shifting and merging together, If nothing else I have created my own world, Once created it will always exist somewhere.

extended primes (black ones)

Edited July 4, 2014 by sunshaker

[Acme]

I am glad I done the Polygonals, It as made me realize that it takes a certain amount of numbers in a sequence to reach 1 or 0.

I agree that the number of elements in the initial condition will have an effect on the ending. So much so in fact that I'm not sure there is much analytic value in these constructions, cool looking as they are.

 

 

All of these 1dimensional cellular automatas are in this xl.

https://alphaomegadotme.files.wordpress.com/2012/05/1-dimensional-cellular-automata-a1.xls

Muchas gracias.

 

 

PS, After doing these yesterday I had a amazing dream where I was in polygonal world, Made up of all these dimensional cellular automatas, all shifting and merging together, If nothing else I have created my own world, Once created it will always exist somewhere.

As I mentioned in the Primes thread, be sure to let out a string as you go so you can find your way back out.

 

On the Polygonal note, here is the sequence of numbers not Polygonal. It contains all the Primes as they are never Polygonal as well as even and odd composites as I already gave earlier.

Sequence of numbers not Polygonal: {7 8 11 13 14 17 19 20 23 26 29 31 32 37 38 41 43 44 47 50 53 56 59 61 62 67 68 71 73 74 77 79 80 83 86 89 97 98 ...}

[sunshaker]

edit, for corrections

Edited July 4, 2014 by sunshaker

[sunshaker]

Noticing a few patterns within prime automata, Cannot show all as it becomes to crowded,

 

 

I start with the centre prime of triangle 165 red, Which i then doubled=330, I then followed to point of triangle(1) then vertically down to prime 383 red, then 383-330=prime 53,all shown with red line,

 

Then moved down 2 places to 157 blue doubled= 314, followed as above to 373-314=prime 59,

 

Then moved down 2 places to 149 black doubled=298, " " to 359-298=prime 61,

 

This works in groups,Shown in different colours, Then there are other patterns within that fills the gaps,

 

 

Hope you can see what I am trying to show, hard for me to explain, Might mean nothing but fun doing.

 

easier to see in xl

https://alphaomegadotme.files.wordpress.com/2012/05/splitting-primes.xls

 

[Acme]

Noticing a few patterns within prime automata, Cannot show all as it becomes to crowded,

 

I start with the centre prime of triangle 165 red, Which i then doubled=330, I then followed to point of triangle(1) then vertically down to prime 383 red, then 383-330=prime 53,all shown with red line,

Then moved down 2 places to 157 blue doubled= 314, followed as above to 373-314=prime 59,

Then moved down 2 places to 149 black doubled=298, " " to 359-298=prime 61,

This works in groups,Shown in different colours, Then there are other patterns within that fills the gaps,

Hope you can see what I am trying to show, hard for me to explain, Might mean nothing but fun doing.

easier to see in xl

You're right; hard to see. As I earlier said, I'm doubtful these mappings 'mean' anything. Hiking the garden path is fun, but know when to turn around and head back.

Did you model the set of not polygonal numbers in post #22? Your post #23 is empty.