Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [triangular-numbers]

Numbers of the form $\sum_{i=1}^ni=1+2+3+4+...+n$.

Triangular numbers are named as such as they represent the number of dots in an equilateral triangle with $n$ dots a side. The sequence (OEIS sequence A000217) has the closed-form $$T_n=\sum\limits_{i=1}^ni=1+2+...+n=\frac{n\cdot(n+1)}2={n+1\choose2}$$

Use this tag solely if your question is about a property concerning this sequence.

33 questions
12
votes
2 answers

Why do the triangular numbers initially form long cycles mod $2^k$?

As discussed at Triangular numbers ($\text{mod } 2^n$) as a permutation of $\{0,1,2,\dots,2^n-1\}$ and What is the set of triangular numbers mod $n$?, mapping the integer $n$ for $0\le n\lt2^k$ to the residue of the corresponding triangular number…
joriki
  • 227,898
  • 14
  • 283
  • 497
6
votes
5 answers

Proving that 2 out of every 3 triangular numbers are divisible by 3

I am trying to prove the observation that the sequence of triangular numbers are divisible in the repeating pattern of not-divisible, divisible and divisible. I've never done proofs before and I'm also a long-time away from doing any maths.…
WarrenTheRabbit
  • 139
  • 7
5
votes
5 answers

Why is it that $1+(1+2+3+4+5+6+\ldots+n)$, basically a triangular number plus $1$, doesn't divide by $3$ or $5$?

As a self learner I am currently learning about triangular numbers, for which the formula is: $$T(n)=\frac{n(n+1)}{2}$$ While playing with my calculator, I added 1 to each resulted number, and I noticed that none of the results divides by $3$ or…
Isaac Brenig
  • 1,385
  • 3
  • 11
5
votes
2 answers

Is there a graphical proof that $9n+1$ is triangular if $n$ is?

It is straightforward to prove that if $n$ is a triangular number then $9n+1$ must be. Is there a systematic decomposition of a triangle made of $9n+1$ dots into nine triangles of $n$ dots plus one left over to illustrate this fact visually?
dbmag9
  • 940
  • 4
  • 12
4
votes
3 answers

Representing the cube of any natural number as a sum of odd numbers

I'm expanding my notes on exercises from Donald Knuth's The Art of Computer Programming, and found something rarely mentioned in the Internet, but still useful to prove Nicomachus' Theorem about the sum of cubes. Knuth phrases this in the following…
Rusurano
  • 546
  • 2
  • 12
3
votes
1 answer

Reciprocal sum of triangular numbers is 1.

n∈N Triangular numbers set is $$\{1,3,6,10,15,21,28,36,45,55,\dots,\frac{n(n+1)}2,\dots\}$$ n≥3 I solved the equation $$\frac 1{x_1} +\frac1{x_2} +\cdots+\frac1{x_n} =1$$ for triangular…
xyz
  • 59
  • 2
3
votes
6 answers

How do I calculate the sum of sum of triangular numbers?

As we know, triangular numbers are a sequence defined by $\frac{n(n+1)}{2}$. And it's first few terms are $1,3,6,10,15...$. Now I want to calculate the sum of the sum of triangular numbers. Let's…
Interstigation
  • 477
  • 2
  • 13
3
votes
0 answers

Has anyone ever studied "triangular" equations?

This is a purely recreational question, just curious if anything out there exists about these. Let $x^\triangle$ denote the triangular number $x(x+1)/2$, then by a triangular equation, I mean an equation of the form $$ax^\triangle+bx+c=0,$$ where…
Luke Collins
  • 8,490
  • 1
  • 17
  • 45
3
votes
2 answers

Integer solutions for $\frac{n(n-1)}2=m^2-1$

I want to find all integer solutions for $$\frac{n(n-1)}2=m^2-1.$$ The only ones I found are $m=\{2,4,64\}$ and $n=\{3,6,91\}$ meaning $m^2-1=\{3,15,4095\}$, but are they the only ones? If not, are there infinitely many solutions? How could I go…
Garmekain
  • 3,102
  • 12
  • 25
3
votes
3 answers

Find sum of consecutive triangular number recursively

In Computer Science, I was asked to write a program that finds the sum of (1 to n)th triangular number, where n is a positive integer. If n=1, result is 1 If n=2, result is (1) + (1 + 2) By testing values from 1 to 8, I found that T(n) = n2 +…
Arkyo
  • 33
  • 4
2
votes
1 answer

Simplifying the alternating sum of n squares

This question is based on a curious problem from Donald Knuth's The Art of Computer Programming, exercise 7 to chapter 1.2.1. It's stated as the following: Formulate and prove by induction a rule for the sums $1^2$, $2^2-1^2$, $3^2-2^2+1^2$,…
Rusurano
  • 546
  • 2
  • 12
2
votes
1 answer

Is there any reasonably standard notation for the triangular root of x?

The triangular number $T_n$ is the total number of objects arranged in a triangle of $n$ rows, i.e. $T_n:=\dfrac{n(n+1)}{2}$. Then, one might ask for the triangular root of a number $x$, as the positive number $n$ (not necessarily an integer) such…
Christopher.L
  • 1,371
  • 1
  • 8
  • 20
2
votes
2 answers

Link between thetrahedral numbers and combinatorics

The triangular numbers are $1, 3=2+1, 6=3+2+1$ and the $n$-th triangular number is $$\binom{n+1}{2}=\frac{n(n+1)}{2}=n+(n-1)+\ldots+2+1.$$ There is a neat explation that the n-te triangular number is $\binom{n+1}{2}$: Consider $n+1$ people. Then…
garondal
  • 602
  • 3
  • 11
2
votes
1 answer

When is the difference of triangular numbers a prime power?

This question is based on Every natural number is covered by consecutive numbers that sum to a prime power. Let $T(n) = \frac{n(n+1)}2$ be the $n$th triangular number, and let $p^j$ denote a prime power. By checking $1 \leq n \leq 10\,000$, it…
Peter Kagey
  • 4,930
  • 10
  • 34
  • 87
2
votes
0 answers

(Why) are $0, 1, 10, 120, 1540, 7140$ the only triangular tetrahedral numbers?

I recently came across entry A027568 of the OEIS, which reports the numbers $$0, 1, 10, 120, 1540, 7140$$ as the set of integers which are both triangular and tetrahedral; that is, each can be written as both $\binom{n+1}{2}$ and $\binom{m+2}{3}$…
Semiclassical
  • 15,277
  • 4
  • 35
  • 86
1
2 3