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groza528
No Place Like Home

 Posted: Sat Jan 12, 2013 9:09 pm    Post subject: 1 What number completes this sequence? 1, 4, 13, 37, 117, __
lostdummy
Daedalian Member

 Posted: Sat Jan 12, 2013 10:44 pm    Post subject: 2 367 ?
groza528
No Place Like Home

 Posted: Tue Jan 15, 2013 5:02 pm    Post subject: 3 Correct, lostdummy Care to explain why? Or leaving it for someone else?
L'lanmal
Daedalian Member

 Posted: Tue Jan 15, 2013 11:16 pm    Post subject: 4 I'm actually very curious now, because these are also the first terms of A091874. However, that series does not terminate after 6 terms like this one does. (It is infinite in fact.) Edit: Nevermind, it is a list of numbers whose squares form a sequence where each successive entry is formed by inserting a single digit into the previous entry. Sequence A091874 allows you to insert more than one digit at a time.
lostdummy
Daedalian Member

Posted: Wed Jan 16, 2013 1:02 pm    Post subject: 5

 groza528 wrote: Correct, lostdummy :-) Care to explain why? Or leaving it for someone else?

Those are roots of these numbers: 1, 16, 169, 1369, 13689, 134689
Each number is square made by inserting single digit into previous number. Presumably you can not extend further with single digits, so it ends with 134689, or its root 367.

BTW, Llanmal posted above link to sequence that extend this with more than one digit, and its 1, 16, 169, 1369, 13689, 134689, 13446889, 1344468889, 134444688889, 13444446888889, ... (just add one 4 and one 8)

But when I tested this, I found another sequence that also satisfy ' 2 digits' condition:
1, 16, 169, 1369, 13689, 134689, 13468900, 1346890000 ... (just add 00 at end)

Guess this could be new addition to On-Line Encyclopedia of Integer Sequences ;p
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