# Finding the Equation of this Sequence

#### newtheory

##### New member
I have what appears to be an infinite sequence:

1, 2, 7, 8, 12, 13, 16, 17, 19, 20, 21, 22, 22, 23, 22, 23, 21, 22, 19, 20, 16, 17, 12, 13, 7, 8, 1, 2, -6, -5, -14, -13...

However, it seems to alternate how it changes, either increasing by one, or increasing by an initial value of five that decreases by one every other number.

I have no clue how I would create an equation for this kind of sequence. This isn't from any textbook, I made the pattern up. If I could get some help creating a sequence, I'd be appreciative.

#### Jomo

##### Elite Member
I have what appears to be an infinite sequence:

1, 2, 7, 8, 12, 13, 16, 17, 19, 20, 21, 22, 22, 23, 22, 23, 21, 22, 19, 20, 16, 17, 12, 13, 7, 8, 1, 2, -6, -5, -14, -13...

However, it seems to alternate how it changes, either increasing by one, or increasing by an initial value of five that decreases by one every other number.

I have no clue how I would create an equation for this kind of sequence. This isn't from any textbook, I made the pattern up. If I could get some help creating a sequence, I'd be appreciative.
Let's see. You made up the pattern and even though you put three dots at the end you do not know if it is an infinite sequence. Then you go onto say If I could get some help creating a sequence, I'd be appreciative. I thought that you know how to make up sequences as you claim to have made up the one you listed. Strange to say the least.

#### tkhunny

##### Moderator
Staff member
I have what appears to be an infinite sequence:

1, 2, 7, 8, 12, 13, 16, 17, 19, 20, 21, 22, 22, 23, 22, 23, 21, 22, 19, 20, 16, 17, 12, 13, 7, 8, 1, 2, -6, -5, -14, -13...

However, it seems to alternate how it changes, either increasing by one, or increasing by an initial value of five that decreases by one every other number.

I have no clue how I would create an equation for this kind of sequence. This isn't from any textbook, I made the pattern up. If I could get some help creating a sequence, I'd be appreciative.
1) ANY finite sequence has infinitely many continuations. It's just the way things are.

2) Since we're just playing, here, look wider. Rather than have each value refer to the just previous value, perhaps it will lead to something if you refer to two previous.

1, 7, 12, 16, 18, etc
2, 8, 13, 17, 20, etc

3) Not all such game sequences are well-motivated.

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#### newtheory

##### New member
Let's see. You made up the pattern and even though you put three dots at the end you do not know if it is an infinite sequence. Then you go onto say If I could get some help creating a sequence, I'd be appreciative. I thought that you know how to make up sequences as you claim to have made up the one you listed. Strange to say the least.
Sorry, for some reason I wrote sequence instead of equation. I know it's an infinite sequence because the rules I applied to the pattern can apply forever.

#### Otis

##### Senior Member
It's kinda peculiar to say both, "I made up a pattern" and "The pattern seems to alternate how it changes". That is, if you created the pattern, then you ought to already know how it changes. :wink:

Not all sequences have a formula, at least not a formula that can predict unlisted elements. However, given a set of known elements in a sequence, we can usually find functions that produce them.

For example, I took the first 16 elements listed and created a system of equations, using a 15-degree polynomial function as the model. For Integer values of x (from 1 through 16), the following function generates the first 16 elements in your sequence. (Disclaimer: I haven't checked my work or the software's result.)

f(x) = -(4/638512875)∙x^15 + (23/28378350)∙x^14 - (1747/36486450)∙x^13 + (97/56700)∙x^12 - (289573/7016625)∙x^11 + (1343/1890)∙x^10 - (32150723/3572100)∙x^9 + (67610519/793800)∙x^8 - (386071927/637875)∙x^7 + (45612233/14175)∙x^6 - (17736090052/1403325)∙x^5 + (1861772146/51975)∙x^4 - (4996868352464/70945875)∙x^3 + (103081963811/1146600)∙x^2 - (2371726369/36036)∙x + 20475

#### Dr.Peterson

##### Elite Member
Sorry, for some reason I wrote sequence instead of equation. I know it's an infinite sequence because the rules I applied to the pattern can apply forever.
How about telling us what rules you applied? Without doing so, you haven't really told us what the sequence is.