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The sequence 1, 1, 1, 21, 3, 1, 4, 1, 5, 1, 6, ... appeared in one of my tutorial sheets in 2019. I assumed that the fourth term, 21, was a mistake (was supposed to be ..., 2, 1, ...) then it made perfect sense to say this must have a formula f(n)=1 when n is odd and f(n)=n/2 when n is even. However, I was told the sequence had no mistake.

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    $\begingroup$ Maybe they didn't want to admit to their mistake :) $\endgroup$ Commented Aug 30, 2020 at 20:57
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    $\begingroup$ Yeah, I think it's pretty obvious that it's supposed to be 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6... $\endgroup$ Commented Aug 30, 2020 at 21:20
  • $\begingroup$ The OEIS (On-line encyclopedia of integer sequences) is the go to source for questions like this and also lends support to the 21 being a 2, 1. $\endgroup$
    – Cardinal
    Commented Aug 31, 2020 at 4:22
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    $\begingroup$ What was the level of the other exercices in the course you were following ? Were they tricky and sophisticated, or simple ? Would 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6... be oddly trivial compared to the other assignments ? $\endgroup$
    – Evargalo
    Commented Nov 19, 2021 at 14:05
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    $\begingroup$ Please read this post to get an idea of why such pattern-finding puzzles are bad. Which is of course not your fault. If there really was no mistake in that tutorial sheet then it is just junk. $\endgroup$
    – user21820
    Commented Nov 19, 2021 at 15:05

1 Answer 1

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maybe 21 wasn't a mistake.

21,91,171,231,351,561,741 etcc are triangular numbers with cadence 1, obtained from the progressive arithmetic sum 1 + 2 + 3 + 4 + 5 + 6 ..... from the sequence it is clear that 21 is obtained from (6 + 1) * (6/2) and that the other numbers are obtained by the formula (x + 1 + k) * (x + k) /2

where x = 6

k=(+7,+5,+3,+5 ) repeted n times

(6+1+7)*(6+7)/ 2 = 91,

(6+1+7+5)*(6+7+5)/2=171

(6+1+7+5+3)*(6+7+5+3)/2 =231

(6+1+7+5+3+5)*(6+7+5+3+5)/2=351

(6+1+7+5+3+5+7)*(6+7+5+3+5+7)/2=561

so the sequency became: ...1,6,1,7,1,8,1,91,10,1,11......1,16,1,171,18 etc

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    $\begingroup$ It looks very arbitrary to me. If I understand well, you are saying that 2,1 becomes 21 because it is a triangular number. OK, but then I would expect 1,5 to merge into 15. And it doesn't. $\endgroup$
    – Florian F
    Commented Nov 19, 2021 at 21:57
  • $\begingroup$ not exactly. I tried to solve the sequence thinking 21 wasn't a mistake, and that the sequence is really 1,1,1,21,3 etc and not 1,1,1,2,1,3 so I don't have to merge 1,5 into 15. furthemore 15 doesn't end with 1, and for the resolution I took the triangolar numbers that end with 1 $\endgroup$
    – user71668
    Commented Nov 20, 2021 at 12:01

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