Discovering formula for a sequence with recurring digits

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Discussion Overview

The discussion revolves around discovering a formula for a sequence characterized by recurring digits, specifically the sequence 1,2,1,2,3,4,1,2,3,4,5,6,1,2,3,4,5,6,7,8,... Participants explore the nature of the sequence, its continuation, and the definition of "digits" versus "numbers." The scope includes mathematical reasoning and conceptual clarification.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants propose that the sequence can be viewed as a nested formula regrouping, with segments like (1,2), (1,2,3,4), etc.
  • Others express confusion about continuing the sequence beyond single digits, questioning the validity of including numbers like 10, 11, and 12 as part of a sequence of recurring digits.
  • Some participants argue that the sequence could include numbers beyond single digits, suggesting a format without commas to represent a continuous sequence of numbers.
  • A participant mentions OEIS (Online Encyclopedia of Integer Sequences) as a resource for such sequences, providing a link for further exploration.
  • One participant attempts to express the sequence using LaTeX notation, indicating a nested sequence structure.
  • Another participant refers to a previous post that includes a formula for the nth term of the series, suggesting it may provide a more detailed answer.

Areas of Agreement / Disagreement

Participants generally do not agree on the interpretation of the sequence as either a sequence of digits or a sequence of numbers. Multiple competing views remain regarding how to define and extend the sequence.

Contextual Notes

There is uncertainty regarding the definitions of "digits" and "numbers," as well as the implications of including multi-digit numbers in the sequence. Some assumptions about the nature of the sequence and its continuation are not fully articulated.

spiritzavior
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please guide me on how to discover the formula for this sequence --> 1,2,1,2,3,4,1,2,3,4,5,6,1,2,3,4,5,6,7,8,...

responses are highly appreciated.
 
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It seems to be a nested formula regrouping we get: (1,2),(1,2,3,4),(1,2,3,4,5,6),(1,2,3,4,5,6,7,8),...
 
spiritzavior said:
please guide me on how to discover the formula for this sequence --> 1,2,1,2,3,4,1,2,3,4,5,6,1,2,3,4,5,6,7,8,...

responses are highly appreciated.
I don't understand how you would continue this sequence as 10,11,12,... are not digits. This makes it improvable that you have a sequence of recurring digits.
 
ramsey2879 said:
I don't understand how you would continue this sequence as 10,11,12,... are not digits. This makes it improvable that you have a sequence of recurring digits.
There is either not enough information or the process repeats modulo 10.
 
ramsey2879 said:
I don't understand how you would continue this sequence as 10,11,12,... are not digits. This makes it improvable that you have a sequence of recurring digits.

Why does it matter that they are not digits? why can't you have:

...1,2,3,4,5,6,7,8,9,10,11,12,1,2,3,4,5,6,7,8,9,10,11,12,13,14...

for example? If you really need a sequence of digits rather than numbers you can just take out the commas:

...1234567891011121234567891011121314...
 
ppnl said:
Why does it matter that they are not digits? why can't you have:

...1,2,3,4,5,6,7,8,9,10,11,12,1,2,3,4,5,6,7,8,9,10,11,12,13,14...

for example? If you really need a sequence of digits rather than numbers you can just take out the commas:

...1234567891011121234567891011121314...
I was confused by the title in which the op called it a sequence with recurring digits rather than a sequence of numbers. It is not a sequence of digits because the 30th term is 10 which is not a digit.
 
my bad, it should be a sequence of numbers not just digits.

what i really mean as a sequence is this 1,2,1,2,3,4,1,2,3,4,5,6,...,1,2,3,4,5,6,7,8,9,10,11,12,1,2,3,4,5,6,7,8,9,10,11,12,13,14,...
 
some more detailed answer/s please.
 
  • #10
[itex]\left\{\left\{k\right\}^{2n}_{k=1}\right\}^{∞}_{n=1}[/itex]

My lack of LaTex knowledge may make this awful, but the idea is a nested sequence, I couldn't find the "s" symbol.
 
  • #11
spiritzavior said:
some more detailed answer/s please.

See the link in the SteveL27 post 6. That link includes a formula for the nth term of the series. I don't think there is a more detailed treatment than that.
 

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