Question

Is there Mathematically any way that you could reproduce a formula for ANY given sequence?

Answer 1

not for ANY but some.

Answer 2

How do you make a formula for any given sequence? Is it just trial-and-error?

Answer 3

actually we see the difference between successive terms, if 'n' th differences are constant, the the function must be polynomial of degree 'n'

Answer 4

without a defining rule, there is no way to know the actual sequence

Answer 5

What if we are given with a sequence which is neither geometric, nor arithmetic, and not even harmonic?

Answer 6

we can form a function for it, but whether or not it is the correct function; or if it converges on the intended sequence for more than just the first known values ... cannot be determined

Answer 7

For example,\[\large \{0,14,78,252\cdots\}\]Can you find the formula for this?

Answer 8

i can, yes

Answer 9

Is there a defined way that you find the formula for ANY sequence?

Answer 10

Let's see how the great Amistre goes about it. :)

Answer 11

you can create a polynomial structure, given 4 knowns, create 4 equations in 4 unknowns to define the coeefs of an x^3 polynomial

Answer 12

You might find this helpful: http://en.wikipedia.org/wiki/Integer_sequence

Extract from that web page:
"An integer sequence is a computable sequence, if there exists an algorithm which given n, calculates an, for all n > 0. An integer sequence is a definable sequence, if there exists some statement P(x) which is true for that integer sequence x and false for all other integer sequences. The set of computable integer sequences and definable integer sequences are both countable, with the computable sequences a proper subset of the definable sequences (in other words, some sequences are definable but not computable). The set of all integer sequences is uncountable (with cardinality equal to that of the continuum); thus, almost all integer sequences are uncomputable and cannot be defined."

Answer 13

Okay, let's give more terms.\[\{ 0,14,78,252,620,1290,2394\cdots\}\]

Answer 14

rref{{1,1,1,1,0},{8,4,2,1,14},{27,9,3,1,78},{64,16,4,1,252}}

10,-35,49,-24

\[f(n)=10n^3-35n^2+49n-24:~n=1,2,3,...\]

Answer 15

the more terms, just means the more equations and the more unknowns to fit the given values to a polynomial

Answer 16

OK, but the original formula behind that is the following:\[\large a_n = n^4 - n\]

Answer 17

then that is one such formula to produce the given values, but without it being defined as such, there is no reason why that is the only one that can produce the given values

Answer 18

What software do you use to produce formula for a given sequence, sir?

Answer 19

the grey matter that rolls about betwixt me ears ...

Answer 20

... I don't understand such creative humor, sir. Your jokes are fabulous but not understandable to the mortals. ;)

Answer 21

;)

Answer 22

im using the wolf at the moment to row reduce a given matrix

Answer 23

i could just as well define a set of slopes across intervals that wold hit the given values and create a linear combination of absolute values that would define it as well

Answer 24

and since the slope between a point can be any number of values, there is right there an infinite number of formulas that can be generated