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Hey guys, I've been into brainteasers since I was little and recently decided to make a puzzle-related website. I figured that this will be the best place to share some of my problems and see if they are any fun. I guess they are too easy, but hope that will entertain you for a bit.

Just another sequence

1, 3, 7, 12, 18, 26, 35, 45, 56, ?

P.S. Please, let me know if you have seen identical puzzle elsewhere, don't want to take someone's credit, even accidentally.

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    $\begingroup$ mathworld.wolfram.com/HofstadterFigure-FigureSequence.html $\endgroup$ – qwertylpc Jul 16 '15 at 19:14
  • $\begingroup$ @qwertylpc Aha! That's the same pattern as I found, only perhaps a more elegant way of expressing it. $\endgroup$ – Rand al'Thor Jul 16 '15 at 19:30
  • $\begingroup$ Thank you guys. As a mathematician, I feel a little bit embarrassed... Should have expected it to be well-known already. $\endgroup$ – Puzzle Prime Jul 16 '15 at 19:32
  • $\begingroup$ @ArturKirkoryan I had a similar experience with my very first question here :-) $\endgroup$ – Rand al'Thor Jul 16 '15 at 19:33
  • $\begingroup$ Yeah, I feel there are not many more fun sequences to be discovered:) $\endgroup$ – Puzzle Prime Jul 16 '15 at 19:35
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We can define the sequence inductively as follows.

  • The first term is $1$.
  • For all $n$, the $(n+1)$th term of the sequence is the sum of the $n$th term and the smallest number which hasn't appeared among the first $n$ terms and the first $n-1$ differences between consecutive terms.

Or in table form:

real terms: $1,3,7,12,18,26,35,45,56,69,\dots$
differences: $2,4,5,6,8,9,10,11,13,14,\dots$

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