Exploring the General Formula of 0 3 5 8 10 13 15 18 20 and Related Sequences

In this article, we will delve into the general formula of the sequence 0 3 5 8 10 13 15 18 20 and explore alternative methods to generate similar sequences. We will also discuss a brute force solution using the J programming language to understand the underlying patterns and formulae.

Understanding the Sequence

The given sequence 0, 3, 5, 8, 10, 13, 15, 18, 20 can be analyzed to identify the underlying pattern. The sequence alternates between two different linear functions depending on the parity of the index n. Let's explore the formulae that generate this sequence.

First Formula

The first formula that produces the given sequence is:

F_n frac{5}{2}n - frac{1}{2} frac{1 - (-1)^{n-1}}{2}

Let's break this down step-by-step:

F_n is the corresponding number in the sequence. n starts at 1 and increases in increments of 1. The term (frac{1 - (-1)^{n-1}}{2}) creates an alternating pattern of 1 and 0, which when scaled by (frac{1}{2}) and subtracted from (frac{5}{2}n - frac{1}{2}), adjusts the sequence accordingly.

Generating the Sequence

To find the 13th number in the sequence, we substitute n 13 into the formula:

F_{13} frac{5}{2} cdot 13 - frac{1}{2} cdot frac{1 - (-1)^{13-1}}{2}

F_{13} frac{5}{2} cdot 13 - frac{1}{2} cdot frac{1 - (-1)^{12}}{2}

F_{13} frac{5}{2} cdot 13 - frac{1}{2} cdot frac{1}{2}

F_{13} frac{5}{2} cdot 13 - frac{1}{4}

Similarly, to find the 14th number:

F_{14} frac{5}{2} cdot 14 - frac{1}{2} cdot frac{1 - (-1)^{14-1}}{2}

F_{14} frac{5}{2} cdot 14 - frac{1}{2} cdot frac{1 - (-1)^{13}}{2}

F_{14} frac{5}{2} cdot 14 - frac{1}{2} cdot frac{1 1}{2}

F_{14} frac{5}{2} cdot 14 - frac{1}{2}

The derived formula ensures that the sequence alternates between two linear increments, giving us:

035810131518202325283033 ...

Alternative Formula

An alternative formula that produces the same sequence is:

F_n frac{5}{2}n - 2 - frac{1}{2} frac{1 - (-1)^n}{2}

This formula is derived by plotting a line through the red points in the sequence plot. Similarly, the term (frac{1 - (-1)^n}{2}) creates an alternating pattern of 1 and 0, and the overall formula ensures the sequence alternates as required.

Brute Force Solution Using J Programming Language

To generate the sequence using the J programming language, we can use the following code:

/10{3 2 3 5 8 10 13 15 18 20 23 25 28 30 33 35 38 40 43 45 48 50

This code generates an alternating pattern 3 2, then computes the running sum to produce the sequence 3, 5, 8, 10, 13, 15, 18, 20, 23, 25, 28, 30, 33, 35, 38, 40, 43, 45, 48, 50.

Conclusion

We have explored two different formulas that produce the sequence 0, 3, 5, 8, 10, 13, 15, 18, 20. Additionally, we have seen how a brute force solution using the J programming language can generate the same sequence by alternating between 3 and 2 and computing the running sum.