Square triangular number
Encyclopedia
In mathematics
, a square triangular number (or triangular square number) is a number which is both a triangular number
and a perfect square
.
There are an infinite
number of square triangular numbers; the first few are 0, 1, 36, 1225, 41616, 1413721, 48024900, 1631432881, 55420693056, 1882672131025 .
The sequences Nk, sk and tk are the OEIS sequences , , and respectively.
In 1778 Leonhard Euler
determined the explicit formula
Other equivalent formulas (obtained by expanding this formula) that may be convenient include
The corresponding explicit formulas for sk and tk are
and
in the following way.
Every triangular number is of the form t(t + 1)/2. Therefore we seek integers t, s such that
With a bit of algebra this becomes
and then letting x = 2t + 1 and y = 2s, we get the Diophantine equation
which is an instance of Pell's equation
. This particular equation is solved by the Pell number
s Pk as
and therefore all solutions are given by
There are many identities about the Pell numbers, and these translate into identities about the square triangular numbers.
s for the square triangular numbers, as well as for the sides of the square and triangle involved. We have
We have
to the continued fraction
for the square root of 2.
A. V. Sylwester gave a short proof that there are an infinity of square triangular numbers, to wit:
If the triangular number n(n+1)/2 is square, then so is the larger triangular number
The generating function for the square triangular numbers is:
and the ratio of successive square triangular numbers approaches 
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a square triangular number (or triangular square number) is a number which is both a triangular number
Triangular number
A triangular number or triangle number numbers the objects that can form an equilateral triangle, as in the diagram on the right. The nth triangle number is the number of dots in a triangle with n dots on a side; it is the sum of the n natural numbers from 1 to n...
and a perfect square
Square number
In mathematics, a square number, sometimes also called a perfect square, is an integer that is the square of an integer; in other words, it is the product of some integer with itself...
.
There are an infinite
Infinity
Infinity is a concept in many fields, most predominantly mathematics and physics, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of infinity...
number of square triangular numbers; the first few are 0, 1, 36, 1225, 41616, 1413721, 48024900, 1631432881, 55420693056, 1882672131025 .
Explicit formulas
Write Nk for the kth square triangular number, and write sk and tk for the sides of the corresponding square and triangle, so thatThe sequences Nk, sk and tk are the OEIS sequences , , and respectively.
In 1778 Leonhard Euler
Leonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...
determined the explicit formula
Other equivalent formulas (obtained by expanding this formula) that may be convenient include
The corresponding explicit formulas for sk and tk are
and
Pell's equation
The problem of finding square triangular numbers reduces to Pell's equationPell's equation
Pell's equation is any Diophantine equation of the formx^2-ny^2=1\,where n is a nonsquare integer. The word Diophantine means that integer values of x and y are sought. Trivially, x = 1 and y = 0 always solve this equation...
in the following way.
Every triangular number is of the form t(t + 1)/2. Therefore we seek integers t, s such that
With a bit of algebra this becomes
and then letting x = 2t + 1 and y = 2s, we get the Diophantine equation
Diophantine equation
In mathematics, a Diophantine equation is an indeterminate polynomial equation that allows the variables to be integers only. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations...
which is an instance of Pell's equation
Pell's equation
Pell's equation is any Diophantine equation of the formx^2-ny^2=1\,where n is a nonsquare integer. The word Diophantine means that integer values of x and y are sought. Trivially, x = 1 and y = 0 always solve this equation...
. This particular equation is solved by the Pell number
Pell number
In mathematics, the Pell numbers are an infinite sequence of integers that have been known since ancient times, the denominators of the closest rational approximations to the square root of 2. This sequence of approximations begins 1/1, 3/2, 7/5, 17/12, and 41/29, so the sequence of Pell numbers...
s Pk as
and therefore all solutions are given by
There are many identities about the Pell numbers, and these translate into identities about the square triangular numbers.
Recurrence relations
There are recurrence relationRecurrence relation
In mathematics, a recurrence relation is an equation that recursively defines a sequence, once one or more initial terms are given: each further term of the sequence is defined as a function of the preceding terms....
s for the square triangular numbers, as well as for the sides of the square and triangle involved. We have
We have
Other characterizations
All square triangular numbers have the form b2c2, where b / c is a convergentConvergent (continued fraction)
A convergent is one of a sequence of values obtained by evaluating successive truncations of a continued fraction The nth convergent is also known as the nth approximant of a continued fraction.-Representation of real numbers:...
to the continued fraction
Continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on...
for the square root of 2.
A. V. Sylwester gave a short proof that there are an infinity of square triangular numbers, to wit:
If the triangular number n(n+1)/2 is square, then so is the larger triangular number
The generating function for the square triangular numbers is:
Numerical data
As k becomes larger, the ratio tk / sk approaches and the ratio of successive square triangular numbers approaches .External links
- Triangular numbers that are also square at cut-the-knotCut-the-knotCut-the-knot is a free, advertisement-funded educational website maintained by Alexander Bogomolny and devoted to popular exposition of many topics in mathematics. The site has won more than 20 awards from scientific and educational publications, including a Scientific American Web Award in 2003,...