Is there a formal name for $ S(k,n) = \sum_{p = 1}^{n} k^p$?

Question

Is there a formal name for

$ S(k,n) = \sum_{p = 1}^{n} k^p$

I tried to use the Online Encyclopedia of Integer Sequences but it returned 11036 results. Also this formula has two inputs and I don't know if/how to do that with OEIS.

TL;DR

This is for use with generating test cases with a computer program. I used Catalan number for generating the test cases for binary operators and am now expanding the generator to handle unary operators.

My first attempt to do this was when ever a unary operator was to be added was to add a binary operator with the second argument set to null, but that resulted in duplicate test cases.

As a check I would like to be able to calculate the number of correct test cases for unary operators using this function; as Catalan number worked correctly for binary operators.

The reason I seek the name is because once I have the name I can query with it and get more helpful web pages, e.g. querying with catalan number binary tree test cases quickly led to Every Binary Tree There Is

Yes I am aware that I will need to combine this with the outcome of the Catalan number, but I am taking this one step at a time.

Edits

Simplification formula noted by Hans Lundmark :

$ S(n,k) = \frac{k(k^n-1)}{(k-1)}$

Answer 1

While there appears to be no formal agreed upon name, OEIS provides a means that can be referenced and includes references to other sources of information.

From OEIS

A228275

$ \begin{array}{c|c|c|c|c|c|c|} & n & 1 & 2 & 3 & 4 & 5 \\ \hline k & & & & & & \\ \hline 1 & & 1 & 2 & 3 & 4 & 5 \\ \hline 2 & & 2 & 6 & 14 & 30 & 62 \\ \hline 3 & & 3 & 12 & 39 & 120 & 363 \\ \hline 4 & & 4 & 20 & 84 & 340 & 1364 \\ \hline 5 & & 5 & 30 & 155 & 780 & 3905 \\ \hline \end{array} $

Column 1: A001477
Column 2: A002378
Column 3: A027444
Column 4: A027445
Column 5: A152031

Row 1: A001477
Row 2: A000918
Row 3: A029858
Row 4: A080674

a(n) is the number of steps which are made when generating all n-step random walks that begin in a given point P on a two-dimensional square lattice. To make one step means to move along one edge on the lattice. - Pawel P. Mazur, Mar 10 2005

This is interesting because Catalan number also references lattice walk.

Row 5: A104891

Answer 2

Maybe the term "sum of the first $n$ numbers of a geomtric progression with initial value $k$ and ratio $k$" is what you are looking for.

Answer 3

This is a geometric series, with two pecularities

• the exponents start at $1$ instead of $0$,

• the would-be-zero-exponent-term is $1$ instead of some arbitrary constant.

As far as I know, there is no specific name for this case, and geometric series is an accurate expression. (Also sum of a geometric progression; it is unclear whether one or the other has a bias towards the infinite summation.)

All search engines will return loads of entries not because the term is vague, but because the topic raises a lot of interest.

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Your way to describe this summation seems to imply that the common ratio $k$ is an integer and that one considers increasing values of $k$. This is not so natural, one usually deals with $$S(a,r,n)=\sum_{i=0}^n ar^i$$ where $a$ and $r$ are abitrary reals.