Find the Pattern for the following family of sequences [closed]

Clash Royale CLAN TAG#URR8PPP

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Can anyone help me find the pattern for an arbitrary sequence of r. For the first family r=1 we have the sequence 1,3,1,3,1,3,... or more easily expressed as [1,3] which repeats indefinitely. For r=2 we have 3,1,7,5,3,1,7,5,... or [3,1,7,5]. Each family is computationally reported as follows:

r=1
[1,3]

r=2
[3,1,7,5]

r=3
[7,13,11,1,15,5,3,9]

r=4
[15,5,3,25,23,13,11,1,31,21,19,9,7,29,27,17]

r=5 [31,53,19,41,39,61,27,49,47,5,35,57,55,13,43,1,63,21,51,9,7,29,59,17,15,37,3,25,23,45,11,33]

r=6 [63,21,115,9,7,29,123,81,79,37,3,25,23,45,11,97,95,53,19,41,39,61,27,113,111,69,35,57,55,77,43,1,127,85,51,73,71,93,59,17,15,101,67,89,87,109,75,33,31,117,83,105,103,125,91,49,47,5,99,121,119,13,107,65]

...

These are numerically derived data points dealing closely with the accelerated qn+1 problem (3n+1 being the Collatz problem). It is easy to generate the first and last points as they are simply [$2^r-1$,..., $2^r+1$] inside a sequence of length $2^r$ but I cannot see a pattern for all the other inside terms.

I tried to graph the sequences in mod($2^r+1$) to try to find a pattern but couldn't see any logical patterns. The blue arrow represents the last entry looping to the first entry (i.e. for r=2, 5 goes back to 3).

Sequences in modular ring of size $2^r+1$

sequences-and-series modular-arithmetic

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edited Jul 27 at 2:51

asked Jul 26 at 19:44

Joseph Schmidt

143

closed as unclear what you're asking by Xander Henderson, Shailesh, Leucippus, amWhy, Adrian Keister Jul 27 at 0:17

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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up vote
1
down vote

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Can anyone help me find the pattern for an arbitrary sequence of r. For the first family r=1 we have the sequence 1,3,1,3,1,3,... or more easily expressed as [1,3] which repeats indefinitely. For r=2 we have 3,1,7,5,3,1,7,5,... or [3,1,7,5]. Each family is computationally reported as follows:

r=1
[1,3]

r=2
[3,1,7,5]

r=3
[7,13,11,1,15,5,3,9]

r=4
[15,5,3,25,23,13,11,1,31,21,19,9,7,29,27,17]

r=5 [31,53,19,41,39,61,27,49,47,5,35,57,55,13,43,1,63,21,51,9,7,29,59,17,15,37,3,25,23,45,11,33]

r=6 [63,21,115,9,7,29,123,81,79,37,3,25,23,45,11,97,95,53,19,41,39,61,27,113,111,69,35,57,55,77,43,1,127,85,51,73,71,93,59,17,15,101,67,89,87,109,75,33,31,117,83,105,103,125,91,49,47,5,99,121,119,13,107,65]

...

These are numerically derived data points dealing closely with the accelerated qn+1 problem (3n+1 being the Collatz problem). It is easy to generate the first and last points as they are simply [$2^r-1$,..., $2^r+1$] inside a sequence of length $2^r$ but I cannot see a pattern for all the other inside terms.

I tried to graph the sequences in mod($2^r+1$) to try to find a pattern but couldn't see any logical patterns. The blue arrow represents the last entry looping to the first entry (i.e. for r=2, 5 goes back to 3).

Sequences in modular ring of size $2^r+1$

sequences-and-series modular-arithmetic

share|cite|improve this question

edited Jul 27 at 2:51

asked Jul 26 at 19:44

Joseph Schmidt

143

closed as unclear what you're asking by Xander Henderson, Shailesh, Leucippus, amWhy, Adrian Keister Jul 27 at 0:17

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

add a comment | 

up vote
1
down vote

favorite

up vote
1
down vote

favorite

Can anyone help me find the pattern for an arbitrary sequence of r. For the first family r=1 we have the sequence 1,3,1,3,1,3,... or more easily expressed as [1,3] which repeats indefinitely. For r=2 we have 3,1,7,5,3,1,7,5,... or [3,1,7,5]. Each family is computationally reported as follows:

r=1
[1,3]

r=2
[3,1,7,5]

r=3
[7,13,11,1,15,5,3,9]

r=4
[15,5,3,25,23,13,11,1,31,21,19,9,7,29,27,17]

r=5 [31,53,19,41,39,61,27,49,47,5,35,57,55,13,43,1,63,21,51,9,7,29,59,17,15,37,3,25,23,45,11,33]

r=6 [63,21,115,9,7,29,123,81,79,37,3,25,23,45,11,97,95,53,19,41,39,61,27,113,111,69,35,57,55,77,43,1,127,85,51,73,71,93,59,17,15,101,67,89,87,109,75,33,31,117,83,105,103,125,91,49,47,5,99,121,119,13,107,65]

...

These are numerically derived data points dealing closely with the accelerated qn+1 problem (3n+1 being the Collatz problem). It is easy to generate the first and last points as they are simply [$2^r-1$,..., $2^r+1$] inside a sequence of length $2^r$ but I cannot see a pattern for all the other inside terms.

I tried to graph the sequences in mod($2^r+1$) to try to find a pattern but couldn't see any logical patterns. The blue arrow represents the last entry looping to the first entry (i.e. for r=2, 5 goes back to 3).

Sequences in modular ring of size $2^r+1$

sequences-and-series modular-arithmetic

share|cite|improve this question

edited Jul 27 at 2:51

asked Jul 26 at 19:44

Joseph Schmidt

143

Can anyone help me find the pattern for an arbitrary sequence of r. For the first family r=1 we have the sequence 1,3,1,3,1,3,... or more easily expressed as [1,3] which repeats indefinitely. For r=2 we have 3,1,7,5,3,1,7,5,... or [3,1,7,5]. Each family is computationally reported as follows:

r=1
[1,3]

r=2
[3,1,7,5]

r=3
[7,13,11,1,15,5,3,9]

r=4
[15,5,3,25,23,13,11,1,31,21,19,9,7,29,27,17]

r=5 [31,53,19,41,39,61,27,49,47,5,35,57,55,13,43,1,63,21,51,9,7,29,59,17,15,37,3,25,23,45,11,33]

r=6 [63,21,115,9,7,29,123,81,79,37,3,25,23,45,11,97,95,53,19,41,39,61,27,113,111,69,35,57,55,77,43,1,127,85,51,73,71,93,59,17,15,101,67,89,87,109,75,33,31,117,83,105,103,125,91,49,47,5,99,121,119,13,107,65]

...

These are numerically derived data points dealing closely with the accelerated qn+1 problem (3n+1 being the Collatz problem). It is easy to generate the first and last points as they are simply [$2^r-1$,..., $2^r+1$] inside a sequence of length $2^r$ but I cannot see a pattern for all the other inside terms.

I tried to graph the sequences in mod($2^r+1$) to try to find a pattern but couldn't see any logical patterns. The blue arrow represents the last entry looping to the first entry (i.e. for r=2, 5 goes back to 3).

Sequences in modular ring of size $2^r+1$

sequences-and-series modular-arithmetic

share|cite|improve this question

edited Jul 27 at 2:51

asked Jul 26 at 19:44

Joseph Schmidt

143

share|cite|improve this question

share|cite|improve this question

edited Jul 27 at 2:51

edited Jul 27 at 2:51

edited Jul 27 at 2:51

asked Jul 26 at 19:44

Joseph Schmidt

143

asked Jul 26 at 19:44

Joseph Schmidt

143

asked Jul 26 at 19:44

Joseph Schmidt

143

143

closed as unclear what you're asking by Xander Henderson, Shailesh, Leucippus, amWhy, Adrian Keister Jul 27 at 0:17

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

closed as unclear what you're asking by Xander Henderson, Shailesh, Leucippus, amWhy, Adrian Keister Jul 27 at 0:17

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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