# What is the formula of the sequence? And how to deduce the formula?

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

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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. –  William Jun 7 '12 at 17:03
Bad Omen: There is nothing with even the first 4 terms in the OEIS. –  Ragib Zaman Jun 7 '12 at 17:04
does it have a formula? Yes. Can we find the formula? No. –  Robert Mastragostino Jun 7 '12 at 17:06
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. –  Robert Mastragostino Jun 7 '12 at 17:32

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 $x_{n+1} = x_n + \Delta x_n$.

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Here is a formula for your sequence

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

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Source? ´ ´ ´ ´ –  Pedro Tamaroff Jun 7 '12 at 17:17
Stephen Wolfram –  user1708 Jun 7 '12 at 17:18
Lol. Interpolation at the values given is not a nice way to find a formula for something ; it doesn't mean anything. –  Patrick Da Silva Jun 7 '12 at 17:20
@PatrickDaSilva Sure. GJB: What is the next number in this number array? $$1,2,3$$ $$4,5,6$$ $$7,8,?$$ –  Pedro Tamaroff Jun 7 '12 at 17:21
@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. :/ –  Pedro Tamaroff Jun 7 '12 at 17:56