some "unknown" sequences

[Joshua Zucker]

On 2/1/06, N. J. A. Sloane <njas at research.att.com> wrote:
> %S A113904 1,7,1,8,9,5
> %A A113904 cynthia (cynthia_tsang87(AT)yahoo.com), Jan 29 2006

This appears as a substring in
  http://www.math.hkbu.edu.hk/UniformDesign/CD2/Level%2011/13_121.txt
but I have no idea what all those numbers are about.  Probably this
has nothing to do with the sequence.

> %S A106648 8,17,32,53
It appears as a substring at
http://www.nap.edu/books/0309045959/html/134.html but I doubt that's
what the author had in mind (especially since the substring isn't even
meaningful, since it crosses one of the boundary lines of the table).

It is a quadratic -- and given that there are four terms, that means
something, right?  If the offset is 0, it's 8 + 9n + 6*n*(n-1)/2, or
in other words 3n^2 + 6n + 8.  So I'm guessing that formula is the
answer to the puzzle.

> %S A113785 2,3,13,175
According to
  http://translate.google.com/translate?hl=en&sl=zh-CN&u=http://iask.sina.com.cn/browse/browse_detail.php%3Fqid%3D3348734&prev=/search%3Fq%3D%25222,3,13,175%2522%26hl%3Den%26hs%3Dov2%26lr%3D%26client%3Dfirefox-a%26rls%3Dorg.mozilla:en-US:official
the answer is:
start with 2, 3.
Thereafter, square previous term plus 2*term before that.
In other words, a(n) = a(n-1)^2 + 2*a(n-2).
In which case, more terms:
2,3,13,175,30651,939484151,882630469980252103,
779036546537560708779146801314890911,
606897940841168991954208317469740424253072728119076070803668647790914127,
368325110597251057525336477431761928758003719261465843398599472365776763423743823713710046939800113576215629993525772418479579862391096917953951

> %S A112027 7,8,4,6,25,26,13,15
I still don't know the answer to this one, but what I do notice is
that the latter four numbers have digit sum equal to the first four
numbers.  Maybe there's something useful going on there.  (By the way,
all these sequence puzzles I don't know the answer to came from
student submissions to Ask Dr. Math, so there's always the possibility
that the students made a typo, or their teachers made a typo, or
things like that)

> %S A112023 1000,72,5,25,7
I don't know the answer to this one either.  Does 7-2 = 5 and 2+5 = 7
have anything to do with it?


here's one more (dumb) puzzle sequences from the same source (student
submissions to Ask Dr. Math, http://mathforum.org/dr.math ), in case
someone feels it worthy of inclusion:
18,9,11,14,7,9,12
Answer (by Floor van Lamoen): repeat divide by 2, add 2, add 3
In which case, all the integer terms are
18 9 11 14 7 9 12 6 8 11
and thereafter you have fractions
11/2, 15/2. 21/2, 21/4, 29/4, 41/4, 41/8, 57/8, 81/8, 81/16, ...

--Joshua Zucker