mathematics

Jacobsthal number

Jacobsthal number is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Jacobsthal number rather than just read about it. In short: In mathematics, the Jacobsthal numbers are an integer sequence named after the German mathematician Ernst Jacobsthal. Like the related Fibonacci numbers, they are a specific type of Lucas sequence U n ( P , Q ) {\displaystyle U_{n}(P,Q)} for which P = 1, and Q = −2—and are defined by a similar recurrence relation: in simple terms, the sequence starts with 0 and 1, then each following number is found by adding the nu…

Key takeaways

Jacobsthal number belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Jacobsthal number to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Jacobsthal number from memory before moving on to harder problems.

Reference excerpt

In mathematics, the Jacobsthal numbers are an integer sequence named after the German mathematician Ernst Jacobsthal. Like the related Fibonacci numbers, they are a specific type of Lucas sequence U n ( P , Q ) {\displaystyle U_{n}(P,Q)} for which P = 1, and Q = −2—and are defined by a similar recurrence relation: in simple terms, the sequence starts with 0 and 1, then each following number is found by adding the number before it to twice the number before that. The first Jacobsthal numbers are:

0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, 683, 1365, 2731, 5461, 10923, 21845, 43691, 87381, 174763, 349525, … (sequence A001045 in the OEIS) A Jacobsthal prime is a Jacobsthal number that is also prime. The first Jacobsthal primes are:

3, 5, 11, 43, 683, 2731, 43691, 174763, 2796203, 715827883, 2932031007403, 768614336404564651, 201487636602438195784363, 845100400152152934331135470251, 56713727820156410577229101238628035243, … (sequence A049883 in the OEIS)

Jacobsthal numbers Jacobsthal numbers are defined by the recurrence relation:

J n = { 0 if n = 0 ; 1 if n = 1 ; J n − 1 + 2 J n − 2 if n > 1. {\displaystyle J_{n}={\begin{cases}0&{\mbox{if }}n=0;\\1&{\mbox{if }}n=1;\\J_{n-1}+2J_{n-2}&{\mbox{if }}n>1.\\\end{cases}}}

The next Jacobsthal number is also given by the recursion formula

J n + 1 = 2 J n + ( − 1 ) n , {\displaystyle J_{n+1}=2J_{n}+(-1)^{n},}

or by

J n + 1 = 2 n − J n . {\displaystyle J_{n+1}=2^{n}-J_{n}.}

The second recursion formula above is also satisfied by the powers of 2. The Jacobsthal number at a specific point in the sequence may be calculated directly using the closed-form equation:

J n = 2 n − ( − 1 ) n 3 . {\displaystyle J_{n}={\frac {2^{n}-(-1)^{n}}{3}}.}

The generating function for the Jacobsthal numbers is

x ( 1 + x ) ( 1 − 2 x ) . {\displaystyle {\frac {x}{(1+x)(1-2x)}}.}

The sum of the reciprocals of the Jacobsthal numbers is approximately 2.7186, slightly larger than e. The Jacobsthal numbers can be extended to negative indices using the recurrence relation or the explicit formula, giving

J − n = ( − 1 ) n + 1 J n / 2 n {\displaystyle J_{-n}=(-1)^{n+1}J_{n}/2^{n}} (see OEIS: A077925) The following identities holds

2 n ( J − n + J n ) = 3 J n 2 {\displaystyle 2^{n}(J_{-n}+J_{n})=3J_{n}^{2}} (see OEIS: A139818)

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Jacobsthal number

Start with the simplest possible case. Write down what Jacobsthal number claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Jacobsthal number before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Jacobsthal number ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Jacobsthal number

In research: Jacobsthal number appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Jacobsthal number in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Jacobsthal number is common in secondary-school and first-year university syllabi. It links to neighbouring topics Integer sequences, Recurrence relations, so understanding it makes those chapters shorter.
In everyday life: Look for Jacobsthal number outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Jacobsthal number in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Jacobsthal number means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Jacobsthal number out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Jacobsthal number in simple terms?

Why does Jacobsthal number matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Jacobsthal number?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Jacobsthal number.