In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a square triangular number (or triangular square number) is a number which is both a
triangular number and a
square number
In mathematics, a square number or perfect square is an integer that is the square (algebra), square of an integer; in other words, it is the multiplication, product of some integer with itself. For example, 9 is a square number, since it equals ...
, in other words, the sum of all integers from
to
has a square root that is an integer. There are
infinitely many square triangular numbers; the first few are:
Solution as a Pell equation
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" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
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 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 the next solution is
Hence there are infinitely many solutions to any Pell equation for which there is one non-trivial one, which is true whenever
is not a square. The first non-trivial solution when
is easy to find: it is
. 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
, is
, and the next, derived from
, is
.
The sequences
,
and
are the
OEIS sequences , , and respectively.
Explicit formula
In 1778
Leonhard Euler
Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...
determined the explicit formula
[
][
]
Other equivalent formulas (obtained by expanding this formula) that may be convenient include
The corresponding explicit formulas for
and
are:
Recurrence relations
The solution to the Pell equation can be expressed as a
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 ...
for the equation's solutions. This can be translated into recurrence equations that directly express the square triangular numbers, as well as 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
, the
square root of 2
The square root of 2 (approximately 1.4142) is the positive real number that, when multiplied by itself or squared, equals the number 2. It may be written as \sqrt or 2^. It is an algebraic number, and therefore not a transcendental number. Te ...
.
[
]
A. V. Sylwester gave a short proof that there are infinitely many square triangular numbers: If the
th triangular number
is square, then so is the larger
th triangular number, since:
The left hand side of this equation is in the form of a triangular number, and as the product of three squares, the right hand side is square.
[
]
The
generating function for the square triangular numbers is:
:
See also
*
Cannonball problem, 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 Union, Soviet-born Israeli Americans, Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow ...
*
Michael Dummett's solution
{{Classes of natural numbers, collapsed
Figurate numbers
Integer sequences