2
$\begingroup$

Does this sequence have a formula?

66 90 117 150 195 264 360 450 540 690 870 1,020 1,260 1,500 1,830 2,160 2,580 3,000 3,510 4,080 4,770 5,490 

If it has, please tell me how to find a formula for this kind of sequence, are there any general ways? thanks

  • 0
    Generally, given a finite amount of a sequence, you can never absolutely tell what the pattern of the sequence is unless you are given additional properties like it is arithmetic. In fact by definition, a sequence of natural numbers is any function from $\omega \rightarrow \omega$. It doesn't have to be a "nice" function.2012-06-07
  • 3
    Bad Omen: There is nothing with even the first 4 terms in the [OEIS](http://oeis.org/search?q=66%2C+90%2C+117%2C+150&sort=&language=english&go=Search).2012-06-07
  • 1
    does it have *a* formula? Yes. Can we find *the* formula? No.2012-06-07
  • 1
    where did you find this from in the first place? It's definitely not obvious; knowing what you're dealing with should narrow the search down.2012-06-07

2 Answers 2

2

I can't tell the answer to your sequence, but I can tell some techniques to find a natural pattern in a finite sequence. First you should understand what it means to "differentiate" finite sequences. The idea I will use will be similar as "integrating the derivative" to find the function.

Define $x_n =$ the $n^{\text{th}}$ term of your sequence and let $\Delta x_n = x_{n+1} - x_n$. Then the sequence $\Delta x_n$ looks like this : $$ 24 \, 27 \, 33 \, 45 \, 69 \, 96 \, 90 \, 90 \, 150 \, 180 \, 150 \, 210 \, 240 \, 330 \, 330 \,... $$ In this case it doesn't help, but it usually does. You can repeat this process until you find a nice pattern (it might not work), and then compute your sequence's formula inductively using $x_{n+1} = x_n + \Delta x_n$.

2

Here is a formula for your sequence

$$27152940-\frac{2036484343487 x}{20748}+\frac{1522831082466208831 x^2}{9777287520}-\frac{109276790067170630629 x^3}{746629228800}+\frac{170957451089886729427 x^4}{1852538688000}-\frac{109373345814921572513 x^5}{2615348736000}+\frac{159374149066660771 x^6}{11208637440}-\frac{31950469591295348797 x^7}{8559323136000}+\frac{4036960149119100781 x^8}{5230697472000}-\frac{335856741144047560741 x^9}{2636271525888000}+\frac{272676794763049 x^{10}}{16094453760}-\frac{70485293749857851 x^{11}}{38626689024000}+\frac{110004747527183 x^{12}}{689762304000}-\frac{1546776215387401 x^{13}}{136949170176000}+\frac{14993028559 x^{14}}{23247544320}-\frac{155106458578751 x^{15}}{5272543051776000}+\frac{11047298359 x^{16}}{10461394944000}-\frac{133065029 x^{17}}{4564972339200}+\frac{454739 x^{18}}{762187345920}-\frac{4147236979 x^{19}}{486580401635328000}+\frac{14639 x^{20}}{193087460966400}-\frac{63307 x^{21}}{200356635967488000}$$

  • 0
    Source? ´ ´ ´ ´2012-06-07
  • 0
    Stephen Wolfram2012-06-07
  • 1
    Lol. Interpolation at the values given is not a nice way to find a formula for something ; it doesn't mean anything.2012-06-07
  • 0
    @PatrickDaSilva Sure. GJB: What is the next number in this number array? $$1,2,3$$ $$4,5,6$$ $$7,8,?$$2012-06-07
  • 0
    @Peter Tamaroff : It doesn't mean that someone makes the mistake of looking arrogant that you have to do the same. Or at least you can do what you just said and say something like "this is just as helpful as what you did ; Wolfram doesn't always give useful answers, you shouldn't rely on it"...2012-06-07
  • 0
    @PatrickDaSilva I wanted the show him the curious sequence above. Can you show why the next term is $27$?2012-06-07
  • 0
    @Peter : Please, don't be arrogant again. I understand your point ; I'm criticizing the manners. We should be respectful to each other on MSE.2012-06-07
  • 0
    @PeterTamaroff Ok, why is it 27?2012-06-07
  • 1
    @PatrickDaSilva I'm not being arrogant. Comments don't convey what people want to say. There has already been a discussion on meta on how "inflexion" can change the meaning of a comment. Again, I'm not being arrongant, I'm trying to show GJB that even the most obvious sequence might be made up into something not obvious. I always try to be respectful. In any case, laughing at an answer isn't. :/2012-06-07
  • 0
    @PatrickDaSilva Your technique only works when the sequence is a polynomial of degree less then terms, this shows its not.2012-06-07
  • 0
    @GJB $$1^2-4\cdot 2 +10=3 $$ $$4^2-4\cdot 5 +10=6 $$ $$7^2-4\cdot 8 +10= 27 $$2012-06-07
  • 0
    Too much Kolmogorov complexity2012-06-07
  • 0
    @GJB : It doesn't only work if the sequence is a polynomial, it might happen that computing the differences just give a sequence that you know, so that you find a formula for it and then "integrate".2012-06-07