In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a square triangular number (or triangular square number) is a number which is both a
triangular number
A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots i ...
and a
perfect square. There are
infinitely many square triangular numbers; the first few are:
:0, 1, 36, , , , , , ,
Explicit formulas
Write for the th square triangular number, and write and for the sides of the corresponding square and triangle, so that
:
Define the ''triangular root'' of a triangular number to be . From this definition and the quadratic formula,
:
Therefore, is triangular ( is an integer)
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bic ...
is square. Consequently, a square number is also triangular if and only if is square, that is, there are numbers and such that . This is an instance of the
Pell equation
Pell's equation, also called the Pell–Fermat equation, is any Diophantine equation of the form x^2 - ny^2 = 1, where ''n'' is a given positive nonsquare integer, and integer solutions are sought for ''x'' and ''y''. In Cartesian coordinates, ...
with . All Pell equations have the trivial solution for any ; this is called the zeroth solution, and indexed as . If denotes the th nontrivial solution to any Pell equation for a particular , it can be shown by the method of descent that
:
Hence there are an infinity of solutions to any Pell equation for which there is one non-trivial one, which holds whenever is not a square. The first non-trivial solution when is easy to find: it is (3,1). A solution to the Pell equation for yields a square triangular number and its square and triangular roots as follows:
:
Hence, the first square triangular number, derived from (3,1), is 1, and the next, derived from , is 36.
The sequences , and are the
OEIS
The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to the ...
sequences , , and respectively.
In 1778
Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries ...
determined the explicit formula
[
][
]
:
Other equivalent formulas (obtained by expanding this formula) that may be convenient include
:
The corresponding explicit formulas for and are:
:
Pell's equation
The problem of finding square triangular numbers reduces to
Pell's equation
Pell's equation, also called the Pell–Fermat equation, is any Diophantine equation of the form x^2 - ny^2 = 1, where ''n'' is a given positive nonsquare integer, and integer solutions are sought for ''x'' and ''y''. In Cartesian coordinates, ...
in the following way.
Every triangular number is of the form . Therefore we seek integers , such that
:
Rearranging, this becomes
:
and then letting and , we get the
Diophantine equation
In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a ...
:
which is an instance of
Pell's equation
Pell's equation, also called the Pell–Fermat equation, is any Diophantine equation of the form x^2 - ny^2 = 1, where ''n'' is a given positive nonsquare integer, and integer solutions are sought for ''x'' and ''y''. In Cartesian coordinates, ...
. This particular equation is solved by the
Pell number
In mathematics, the Pell numbers are an infinite sequence of integers, known since ancient times, that comprise the denominators of the closest rational approximations to the square root of 2. This sequence of approximations begins , , , , an ...
s 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 relation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
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 , where is a
convergent to the
continued fraction expansion of
.
[
]
A. V. Sylwester gave a short proof that there are an infinity of square triangular numbers:
[
] If the th triangular number is square, then so is the larger th triangular number, since:
:
As the product of three squares, the right hand side is square. The triangular roots are alternately simultaneously one less than a square and twice a square if is even, and simultaneously a square and one less than twice a square if is odd. Thus,
:49 = 7
2 = 2 × 5
2 − 1,
:288 = 17
2 − 1 = 2 × 12
2, and
:1681 = 41
2 = 2 × 29
2 − 1.
In each case, the two square roots involved multiply to give : , , and .
Additionally:
:
, , and . In other words, the difference between two consecutive square triangular numbers is the square root of another square triangular number.
The
generating function
In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary serie ...
for the square triangular numbers is:
:
Numerical data
As becomes larger, the ratio approaches
≈ , and the ratio of successive square triangular numbers approaches ≈ . The table below shows values of between 0 and 11, which comprehend all square triangular numbers up to .
:
See also
*
Cannonball problem
In the mathematics of figurate numbers, the cannonball problem asks which numbers are both square and square pyramidal. The problem can be stated as: given a square arrangement of cannonballs, for what size squares can these cannonballs also be a ...
, on numbers that are simultaneously square and square pyramidal
*
Sixth power, numbers that are simultaneously square and cubical
Notes
External links
Triangular numbers that are also squareat
cut-the-knot
Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Math ...
*
Michael Dummett's solution
{{Classes of natural numbers, collapsed
Figurate numbers
Integer sequences