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What Is a Triangular Number?
A Mathematical Problem of Intersections
This mathematical question was asked of Hubbers:
Fibonacci numbers are numbers of the form:
1,1, 2, 3, 5, 8, 13,21, 34, 55,89...x, y, x+y.......
while triangular numbers are numbers of the form:
1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66...
Now my question: Using an equation can we show a relationship between Fibonacci Numbers and Triangular Numbers? If yes, what is it?
The short answer is that there is an intersection of the two formulas that yields Pascal's Triangle.
How do we derive a formula for Pascal's Triangle?
I think there is probably at least one intersection that can be found between the two number sequences in the question that would aide in gaining a formula. Finding the formula is rather complex.When thinking about this dilemma for the first time, one might see images of intersections of conic sections or a straight addition of two formulas to find a crossingpoint, or even some sort of fractal arrangement. It can be a complex matter in which an investigator gets in his own way during the derivation of a formula. However, with enough time to think about the problem, the exercise can be entertaining.
Triangular
Click thumbnail to view fullsizeNumber Theory
Trangles and Quadrilaterals
This is fun for individuals that like number puzzles and/or geometry.
Triangular numbers are made up of equilateral triangular arrangements and Fibonacci numbers are built on spiraled squares and rectangles.Since equilateral triangles can fit into squares and rectangles in certain instances as to fill up the area of those quadrilaterals, there would seem to be a relationship between the 3 and 4sided figures.
An intuitive assumption is that if some number and size of equilateral triangles can fit completely into the specific squares and rectangles observed, then a relationship between triangular numbers and Fibonacci numbers exists. Some equilateral triangles may need to be bisected and the two parts realigned around another triangle for this to be so in some cases. Some other division of the triangles may be necessary in other cases. We see this type of problem in books of puzzles frequently.
The formula for Triangular Numbers is present in a variety of text books:
Formula for Triangular Numbers (T)
TABLE I. Examples of Triangular Numbers
n
 n (n+1)/2
 T


1
 1(2)/2
 1

2
 2(3)/2
 3

3
 3(4)/2
 6

4
 4(5)/2
 10

5
 5(6)/2
 15

6
 6(7)/2
 21

7
 7(8)/2
 28

8
 8(9)/2
 36

9
 9(10)/2
 45

10
 10(11)/2
 55

20
 20(21)/2
 210

Graphically, a series of triangular numbers looks like a line of twodimensional pyramids becoming larger form left to right.
For fun, let's look at a graph of the triangular numbers themselves.
GRAPH I. Triangular Numbers Graph from TABLE I.
Fibonacci Sequence Numbers and Formula
Every Fibonacci Number in this series is the sum of the preceding two numbers. This formula is also logged in many mathematics texts and handbooks:
F_{n} = F_{n1} + F_{n2 }
or_{}
Fibonacci Numbers Expression
Binet's Formula for the "nth" Fibonacci Number in a Sequence
To easily find the "nth" number in a Fibonacci Sequence, use this online calculator.
 Fibonacci Calculator
Fibonacci numbers and the golden section calculator with multiprecision arithmetic. Also powers of Phi and the Rabbit sequence.
TABLE II. A Fibonacci Series
Sum of Two Previous


1

1

2

3

5

8

13

21

34

GRAPH II. Graph of Fibonnacci Numbers from TABLE II.
A sequence of Fibonacci Numbers ultimately leads to a spiral(ing) shape, one that encompasses a fractal equation(s). Enjoy the animation and the video below for more answers.
Fibonacci Numbers
Musical Fibonacci Sequence
A Fibonacci Spiral
Experiment With Numbers
As an experiment, we might try simply adding together the two equations  adding the resulting points together. It will likely end up a nonsense curve when graphed. Nevertheless, someone will try it to see what it looks like.
TABLE III. An Experimental Addition
T
 Fibonacci
 Result


3
 1
 4

6
 1
 7

10
 2
 12

15
 3
 18

21
 5
 26

28
 8
 36

36
 13
 49

45
 21
 66

GRAPH III. Experimental Graph From TABLE III.
ADDING EQUATIONS
The other thing to try is to add the formulas for nth Triangular Number and nth Fibonacci Number. This is quite tedious. but can be done.
(n^{2} + n)/2
plus, rounding the square root of 5 to 2.24 in Binet's Formula we can simplify:
(5.24^{n}  0.77^{n})/2.24
So the addition of formulas is:
(n^{2} + n)/2 + (5.24^{n}  0.77^{n})/2.24
TABLE IV. Addition of T and F Equations.
n
 T
 F
 Result


0
 0.0
 0.0
 0.0

1
 0.5
 2.03
 2.53

2
 3.0
 12.02
 15.02

3
 6.0
 64.03
 67.03

4
 10
 336.45
 345.45

GRAPH IV. Graph from TABLE IV.
A Triangle Calculator
A the end of experiments and derivations, mathematics has found that a relationship between triangular number and Fibonacci numbers exists in Pascal's Triangle. The information below reveals construction, formulas, and associated uses for the specialized triangle that is like a calculator.
Before Pascal's Triangle
The above diagram is a triangle from Ancient China that was used to assist in finding cube roots of numbers. It is Pascal's Triangle before Blaise Pascal discovered it in the 13th C. The triangle, also found in other ancient societies like Japan, Persia, and India, is an instance of an intersection of or relationship between triangular numbers and Fibonacci numbers.
Pascal's Formula
A Pascal's Triangle Online Calculator
 Pascal's Triangle Calculator and Others
Pascal Triangle Calculator, binomial coefficient calculator,Floyd Triangle Calculator,Leibniz Harmonic Triangle Calculator  Dr. Math Presents Pascal's Triangle and Its Applications
What is Pascal's Triangle? How do you construct it? What is it used for? Find a formula here.
A Pascal's Triangle
Triangular numbers are contained within Pascal's Triangle. Start with the first "1" in the second row from the top of the triangle and look diagonally down through a line moving to the southeast. The larger the triangle you create, the greater the number of triangular numbers you will be able to read.
Pascal's Triangle can be used for probabilities and for other calculations (see videos below).
Use Pascal's Triangle to Figure Binomial Coefficients
Related History
A Fractal Pattern within Pascal's Triangle
"Magic" Properties of Pascal's Triangle.
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© 2011 Patty Inglish
Comments
Amazing! This is very interesting and informative.This hub helps to bring out the beauty, harmony, elegance, aesthetics inherent in mathematics. Number theory is an amazing field of study.
Keep up the good work.
2 + 2 = 5
... for very large values of 2.
Hello Patty, this is way above me, I only had elemantary schooling and commercial college.
Amazing hub. I had wanted two mathematical expressions A and B that use Fibonacci numbers as inputs to generate evensubscripted and oddsubscripted triangular numbers respectively.
I remember Pascal's Triangle...probably one of the most useful tools I have ever remembered in math. (Besides the trigonometric circle and all its bits and pieces) Great hub! Voted UP :)
Well, l`pressed up and awesome. because your hubs always are, but l´m afraid l was lost before halfway down the page... Just not my area, l think.
Oh, wow! I'm rather averse to math, but I loved this Hub! Very well explained and illustrated!
Great way to simplify concept.this is a real model hub!
Fun stuff, outstanding answer to the question What Is a Triangular Number?
Thanks for all the information. Mike
That's very interesting. I always love math.
Dearest Patty,
Ahhhhhh, now I know how you produce all these amazing Hubs! You have an entire staff of brilliant and genius researchers which you pay millions!! ;)
I'm not even particularly fond of numbers/mathematics and you had me reading each and every entry with fascination and amazement!
This Hub is its own Fibonacci Spiral! And you are the amazing Intersection of all!
Blessings always, Earth Angel!
Interesting Hub. Thank you for sharing.
Hi Patty,an interesting article.
Although I was unaware of them being triangular numbers, I use them to calculate the total combinations I have to make to cover all of a selection of numbers for lotto, (or horses in a race) An exercise, I'm not much of a gambler.
As an example, if I have to combine 5 numbers of 8 non sequentially every possible way, the action goes: 1+2+3+4+5, 1+2+3+4+6, 1+2+3+4+7, 1+2+3+4+8, (total 4) 1+3+4+5+6, 1+3+4+5+7, 1+3+4+5+8 (total 3) 1+4+5+6+7, 1+4+5+6+8 (total 2) and 1+5+6+7+8 (total 1) giving a sub total for the 1 series of 4+3+2+1=10 combinations. Dropping the 1 and doing the 2 sequence gives 104=6 combinations the next 63=3,and the4 sequence 32=1 (4+5+6+7+8, see told you so!)
So the total combo's are 10+6+3+1=20. Does this sequence look familiar? yep, triangular.
It's a lot easier and quicker to do than the tedious explanation, but I'll give that bit a miss. Even the shortcut is a pain to explain, thanks to my convoluted mathematical brain.
Cheers
Peter.
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