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I often have to generate a polynomial sequence so I created this simple online utility that does it for me. It lets you generate however many polynomial series numbers you need in any base. It works in the browser and is powered by alien technology from the future.

## Polynomial Progression Generator Options

Fixed Value and Ratio
Count and Separator
Base

## Polynomial Progression Generator Examples (click to try!)

Positive Polynomial Series
In this example, we generate an increasing polynomial sequence with a positive fixed value of 2 and a positive common ratio of 3. It is calculated by the formula pₙ = a + bⁿ, where pₙ – nth sequence number, a – constant fixed value, b – common ratio. By definition, we set b0 to be 0 and therefore we have the following series – p₀ = 2, p₁ = 2 + 3¹, p₂ = 2 + 3², p₃ = 2 + 3³, p₄ = 2 + 3⁴, …. We generate the series up to p₂₀ and separate polynomial terms by commas.
`2, 5, 11, 29, 83, 245, 731, 2189, 6563, 19685, 59051, 177149, 531443, 1594325, 4782971, 14348909, 43046723, 129140165, 387420491, 1162261469`
Series fixed value.
Series common ratio.
How many values to generate?
Separate terms with this value.
Decimal Base
Constant Terms
In this example, we set the common ratio equal to zero (b = 0). As a result, our sequence becomes p₀ = a, p₁ = a + 0¹, p₂ = a + 0², p₃ = a + 0³, … (where a is the fixed value). This series has constant terms all equal to "a", which in this example is set to 5. It's also the same as a constant arithmetic progression (with the difference of 0).
`5 5 5 5 5 5 5 5 5 5 5 5 5 5 5`
Series fixed value.
Series common ratio.
How many values to generate?
Separate terms with this value.
Decimal Base
Geometric Progression
This example generates polynomial terms without the fixed value (a = 0). In this case, the polynomial formula has the form pₙ = bⁿ. It's the same as a geometric progression gₙ = c×bⁿ⁻¹ with c = 1 however, in polynomial progression, there is also a zero value p₀ = 0.
`0; 4; 16; 64; 256; 1024; 4096; 16384; 65536; 262144`
Series fixed value.
Series common ratio.
How many values to generate?
Separate terms with this value.
Decimal Base
Negative Fixed Value
This example uses a large negative fixed value (a = -256) and a small positive common ratio (b = 2). As b > 1, the last term bⁿ in the formula pₙ = a + bⁿ grows very quickly because it's exponential. As a result, the positive common ratio term quickly outweighs the negative fixed value and after just 9 steps, the elements became positive.
```-256
-254
-252
-248
-240
-224
-192
-128
0
256
768
1792```
Series fixed value.
Series common ratio.
How many values to generate?
Separate terms with this value.
Decimal Base
Decimal Polynomial Series
This example uses a tiny decimal fraction as the common ratio equal to 0.1. Exponential functions (such as bⁿ) with small positive bases (0 < b < 1) converge to 0 when n increases. In the polynomial formula pₙ = a + bⁿ we set a = 1, b = 0.1 and it becomes pₙ = 1 + 0.1ⁿ.
```1
1.1
1.01
1.001
1.0001
1.00001
1.000001
1.0000001
1.00000001
1.000000001```
Series fixed value.
Series common ratio.
How many values to generate?
Separate terms with this value.
Decimal Base
Cyclic Polynomial Sequence
In this example, we set the common ratio value equal to a negative number -1. When it's raised to an even power, we get 1, and when it's raised to an odd power, we get -1. We use the constant 2 for the initial term and get the sequence 2, 1, 3, 1, 3, 1, …. As you can see, it becomes cyclic from the second term. When it's converted to the binary base, it becomes 10, 1, 11, 1, 11, 1, …. For the output, the separator is set to a new line symbol and each binary term is printed on its own line.
```10
1
11
1
11
1
11
1
11
1```
Series fixed value.
Series common ratio.
How many values to generate?
Separate terms with this value.
Binary Base

## How Does This Polynomial Progression Generator Work?

This polynomial series generator works entirely in your browser and is written in JavaScript. It implements a sequence generator that uses the polynomial formula pₙ = a + bⁿ, where `pₙ` is the n-th sequence term, `a` is the fixed constant value (can be specified in options), and `b` is the common-ratio (also can be changed in options). It uses the variable name `fixedValue` for `a` and the variable name `commonRatio` for `b`. Both values are created from strings by constructing `new BigNumber()` objects. It uses an array `polynomials` to store sequence elements and adds the first `fixedValue` element to it via `var polynomials = [fixedValue]`. Then, it runs a `for` loop `count` times (given in options), with the start value `var i = 1` and an increment of 1 (via `i++`). It calculates the polynomial formula via two BigNumber library methods. The first one, `pow()`, raises `commonRatio` to the power `i` and the second one, `plus()`, adds `fixedValue` to the number. Then it converts the polynomial term to the selected base via `toString(base)` function and `push()`es it to the `polynomials` array. When the loop finishes, all values are joined together via `join(sep)` function, where `sep` is the output separator (specified in options).

### Created by Browserling

This polynomial progression generator was created by me and my team at Browserling. Behind the scenes, it's actually powered by our web developer tools that are used by millions of people every month. Browserling itself is an online cross-browser testing service powered by alien technology. Check it out!

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