Lozanić's triangle

Key takeaways

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

Reference excerpt

Lozanić's triangle (sometimes called Losanitsch's triangle) is a triangular array related to of Pascal's triangle. It is named after the Serbian chemist Sima Lozanić, who researched it in his investigation into the symmetries exhibited by rows of paraffins (archaic term for alkanes) and isomer types and number of alkanes. First 9 rows of the triangle (sequence A034851 in the OEIS) with their row sums 2 n − 2 + 2 ⌊ n / 2 ⌋ − 1 {\displaystyle 2^{n-2}+2^{\lfloor n/2\rfloor -1}} (OEIS: A005418).

k 0 1 2 3 4 5 6 7 8 sum n 0 1 1 1 1 1 2 2 1 1 1 3 3 1 2 2 1 6 4 1 2 4 2 1 10 5 1 3 6 6 3 1 20 6 1 3 9 10 9 3 1 36 7 1 4 12 19 19 12 4 1 72 8 1 4 16 28 38 28 16 4 1 136

As in Pascal's triangle, the outer diagonals are all 1s. Most of the enclosed numbers are the sum of the two numbers above. But for even n and odd k one also needs to subtract the number at position ( n 2 − 1 , k − 1 2 ) {\displaystyle \left({\frac {n}{2}}-1,{\frac {k-1}{2}}\right)} of Pascal's triangle. The diagonals next to the edge diagonals contain the positive integers in order, but with each integer stated twice (OEIS: A004526). Moving inwards, the next pair of diagonals contain the "quarter-squares" (OEIS: A002620), or the square numbers and pronic numbers interleaved. The next pair of diagonals contain the alkane numbers l(6, n) (OEIS: A005993). And the next pair of diagonals contain the alkane numbers l(7, n) (OEIS: A005994), while the next pair has the alkane numbers l(8, n) (OEIS: A005995), then alkane numbers l(9, n) (OEIS: A018210), then l(10, n) (OEIS: A018211), l(11, n) (OEIS: A018212), l(12, n) (OEIS: A018213), etc. The sums of the diagonals of Lozanić's triangle intermix F 2 n − 1 + F n + 1 2 {\displaystyle {F_{2n-1}+F_{n+1}} \over 2} with F 2 n + F n 2 {\displaystyle {F_{2n}+F_{n}} \over 2} (where Fx is the xth Fibonacci number). The illustrations below show five related number triangles. Entry (n, k) in Pascal's triangle (blue) is the number of binary strings of length n and weight k. The respective entry of Lozanić's triangle (green) counts these strings up to reversal, i.e. treating strings symmetric to each other as equivalent. Another triangle (red, OEIS: A051159) counts the palindromic binary strings. The difference of Pascal's and Lozanić's triangle (dark brown, OEIS: A034852) counts the pairs of chiral strings. (It has applications in the chemical study of catacondensed polygonal systems.)

References S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917 - 1926. N. J. A. Sloane, Classic Sequences

Illustrations

Worked examples

Example 1 — a first encounter with Lozanić's triangle

Start with the simplest possible case. Write down what Lozanić's triangle 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 Lozanić's triangle 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 Lozanić's triangle 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 Lozanić's triangle

In research: Lozanić's triangle 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 Lozanić's triangle 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: Lozanić's triangle is common in secondary-school and first-year university syllabi. It links to neighbouring topics Factorial and binomial topics, Triangles named after people, Triangles of numbers, so understanding it makes those chapters shorter.
In everyday life: Look for Lozanić's triangle 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.

How to study Lozanić's triangle in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Lozanić's triangle 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 Lozanić's triangle out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Lozanić's triangle in simple terms?

Lozanić's triangle (sometimes called Losanitsch's triangle) is a triangular array related to of Pascal's triangle. It is named after the Serbian chemist Sima Lozanić, who researched it in his investigation into the symmetries exhibited by rows of paraffins (archaic term for alkanes) and isomer type…

Why does Lozanić's triangle 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 Lozanić's triangle?

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 Lozanić's triangle.