The discussion revolves around identifying the next number in the sequence: 1, 1, 3, 6, 18, ? Participants explore various mathematical approaches and interpretations of what constitutes a "logical" continuation of the sequence.
Participants do not reach a consensus on the next number in the sequence. Multiple competing views and methods are presented, with no single approach being universally accepted.
Participants note the complexity of the problem, with some expressing frustration over the lack of a clear, simple pattern. The discussion highlights the dependence on definitions of "logical" and the assumptions underlying different mathematical models.
The next number could literally be anything. We need some kind of reference. What are you working on right now?lfdahl said:What is the next logical number in the sequence:
1, 1, 3, 6, 18, ??
lfdahl said:What is the next logical number in the sequence:
1, 1, 3, 6, 18, ??
The OEIS lists several possibilities, the simplest of which is the sequence given by $a_n = a_{n-1} + (n-1)a_{n-2}$, with $a_1=a_2=1$. The next term is then $a_6 = 48$.lfdahl said:What is the next logical number in the sequence:
1, 1, 3, 6, 18, ??
anemone said:I would say, given the first two terms as $1, 1$, the sequence
[TABLE="class: grid, width: 500"]
[TR]
[TD]n=1[/TD]
[TD]n=2[/TD]
[TD]n=3[/TD]
[TD]n=4[/TD]
[TD]n=5[/TD]
[TD]n=6[/TD]
[/TR]
[TR]
[TD]1[/TD]
[TD]1[/TD]
[TD]3[/TD]
[TD]6[/TD]
[TD]18[/TD]
[TD]-[/TD]
[/TR]
[/TABLE]
can be viewed as
[TABLE="class: grid, width: 500"]
[TR]
[TD]n=1[/TD]
[TD]n=2[/TD]
[TD]n=3[/TD]
[TD]n=4[/TD]
[TD]n=5[/TD]
[TD]n=6[/TD]
[/TR]
[TR]
[TD]1[/TD]
[TD]1[/TD]
[TD]$1+1+1^0=3$[/TD]
[TD]$1+3+2^1=6$[/TD]
[TD]$3+6+3^2=18$[/TD]
[TD]-[/TD]
[/TR]
[/TABLE]
i.e., for $n>2$, $a_n=a_{n-2}+a_{n-1}+ (n-2)^{n-3}$.
Hence, the next number $(a_6)$ would be $a_6=a_4+a_5+(6-2)^{6-3}=6+18+4^3=88$.
Does this look right, Dan?(Wondering)
Thankyou very much! I think the simpler the solution the better.Opalg said:The OEIS lists several possibilities, the simplest of which is the sequence given by $a_n = a_{n-1} + (n-1)a_{n-2}$, with $a_1=a_2=1$. The next term is then $a_6 = 48$.
Then there's the "classical" fit scheme: The given numbers specify a 4th degree polynomial (or higher):lfdahl said:What is the next logical number in the sequence:
1, 1, 3, 6, 18, ??
Yes, you´re absolutely right. I expected a quite simple explanation far from fifth degree polynomials etc. This was my purpose of asking using the term "logical".HallsofIvy said:Then what was your purpose in asking this question? What do you mean by "logical number"? There are, as topsquark said, an infinite number of perfectly "logical" formulas that will give those five numbers and then whatever sixth number you want.
In fact, you can assign any number you want as the sixth number and then find the unique fifth degree polynomial that will give those six numbers.
topsquark said:Then there's the "classical" fit scheme: The given numbers specify a 4th degree polynomial (or higher):
[math]\frac{3}{8}x^4 - \frac{47}{12} x^3 + \frac{121}{8} x^2 - \frac{283}{12} x + 13[/math]
This gives the next number as 56.
We can "force" the next number to be anything we want. For example:
[math]- \frac{37}{120} x^5 + 5x^4 - \frac{241}{8} x^3 + \frac{169}{2} x^2 - \frac{1621}{15} x + 50[/math]
gives the next number as 19.
Or you can add logarithmic terms, trig terms, etc. (The coefficients will be really messy, but it can be done.)
-Dan
PS Fond gratitude to Mathematica for slugging through the simultaneous equations!