I wrote a program that basicly can find the formula of the sequence that created with any-degree equation. For example if you give my program that sequence: [1926, 2811, 833240, 28778265, 398155842, 3267499271, 18997669476, 86360995365, 326275263710, 1066959810387, 3107266779216, 8227981408241, 20122182489690, 45999898361055] My program is going to return you a equation immediately.
There is no problem that kind of sequences: 13*x^11 + -22*x^10 + -15*x^9 + 20*x^8 + -5*x^7 + 13*x^6 + -33*x^5 + 11*x^4 + -6*x^3 + 54*x^2 + -101*x + 1997
I want to do same thing in geometric sequences but fail. For example:
I have a simple geometric sequence like that: (2^x) [2, 4, 8, 16, 32, 64, 128, 256, 512] And another one: (3^x) [3, 9, 27, 81, 243, 729, 2187, 6561, 19683]
I can find nth term of these simple sequences and i can create rest of sequence. I know there is a formula : a*d^n-1
There is no problem until now. But what if add these number and i create a new sequence.
For example: (3^x + 2^x + 1)
[6, 14, 36, 98, 276, 794, 2316, 6818, 20196]
I can't get any relationship.
My question : "Is there a relationship between these numbers?" If there is, how can i find it? Unless, why?
As you have already said: you have a program to find the rule to a sequence. If I give you two numbers, there is a unique linear sequence with those two numbers as the first two terms. Say $a_1=5$ and $a_2 = 7$ then the linear sequence with $a_n = 2n+5$ has $a_1 = 5$ and $a_2 = 7$. But what about the quadratic sequence with $a_n = kn^2 + (2-3k)n + (2k+3)$? This sequence has $a_1 = 5$ and $a_2 = 7$ for all possible choices of $k$. There is an infinite, uncountable number of quadratic sequences with $a_1 = 5$ and $a_2 = 7$. I could find an order ten sequence which has the same first nine terms of $3^n+2^n+1$. There is an infinite, uncountable number of order 11 sequence which have the same first nine terms of $3^n+2^n+1$.