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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 1 to n 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 N_k for the kth square triangular number, and write s_k and t_k for the sides of the corresponding square and triangle, so that Define the ''triangular root'' of a triangular number N=\tfrac to be n. From this definition and the quadratic formula, Therefore, N is triangular (n 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 ...
8N+1 is square. Consequently, a square number M^2 is also triangular if and only if 8M^2+1 is square, that is, there are numbers x and y such that x^2-8y^2=1. This is an instance of the Pell equation x^2-ny^2=1 with n=8. All Pell equations have the trivial solution x=1,y=0 for any n; this is called the zeroth solution, and indexed as (x_0,y_0)=(1,0). If (x_k,y_k) denotes the kth nontrivial solution to any Pell equation for a particular n, 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 n is not a square. The first non-trivial solution when n=8 is easy to find: it is (3,1). A solution (x_k,y_k) to the Pell equation for n=8 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 6\cdot (3,1)-(1,0)-(17,6), is 36. The sequences N_k, s_k and t_k 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 s_k and t_k 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 b^2c^2, where \tfrac is a convergent to the continued fraction expansion of \sqrt2, 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 nth triangular number \tfrac is square, then so is the larger 4n(n+1)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: :\frac = 1 + 36z + 1225 z^2 + \cdots


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 square
at
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 ...
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Michael Dummett's solution
{{Classes of natural numbers, collapsed Figurate numbers Integer sequences