In mathematics, a square triangular number (or triangular square number) is a number which is both a triangular number and a square number. There are infinitely many square triangular numbers; the first few are:
Write N k {\displaystyle N_{k}} for the k {\displaystyle k} th square triangular number, and write s k {\displaystyle s_{k}} and t k {\displaystyle t_{k}} for the sides of the corresponding square and triangle, so that
Define the triangular root of a triangular number N = n ( n + 1 ) 2 {\displaystyle N={\tfrac {n(n+1)}{2}}} to be n {\displaystyle n} . From this definition and the quadratic formula,
Therefore, N {\displaystyle N} is triangular ( n {\displaystyle n} is an integer) if and only if 8 N + 1 {\displaystyle 8N+1} is square. Consequently, a square number M 2 {\displaystyle M^{2}} is also triangular if and only if 8 M 2 + 1 {\displaystyle 8M^{2}+1} is square, that is, there are numbers x {\displaystyle x} and y {\displaystyle y} such that x 2 − − --> 8 y 2 = 1 {\displaystyle x^{2}-8y^{2}=1} . This is an instance of the Pell equation x 2 − − --> n y 2 = 1 {\displaystyle x^{2}-ny^{2}=1} with n = 8 {\displaystyle n=8} . All Pell equations have the trivial solution x = 1 , y = 0 {\displaystyle x=1,y=0} for any n {\displaystyle n} ; this is called the zeroth solution, and indexed as ( x 0 , y 0 ) = ( 1 , 0 ) {\displaystyle (x_{0},y_{0})=(1,0)} . If ( x k , y k ) {\displaystyle (x_{k},y_{k})} denotes the k {\displaystyle k} th nontrivial solution to any Pell equation for a particular n {\displaystyle 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 {\displaystyle n} is not a square. The first non-trivial solution when n = 8 {\displaystyle n=8} is easy to find: it is ( 3 , 1 ) {\displaystyle (3,1)} . A solution ( x k , y k ) {\displaystyle (x_{k},y_{k})} to the Pell equation for n = 8 {\displaystyle 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 ) {\displaystyle (3,1)} , is 1 {\displaystyle 1} , and the next, derived from 6 ⋅ ⋅ --> ( 3 , 1 ) − − --> ( 1 , 0 ) − − --> ( 17 , 6 ) {\displaystyle 6\cdot (3,1)-(1,0)-(17,6)} , is 36 {\displaystyle 36} .
The sequences N k {\displaystyle N_{k}} , s k {\displaystyle s_{k}} and t k {\displaystyle t_{k}} are the OEIS sequences OEIS: A001110, OEIS: A001109, and OEIS: A001108 respectively.
In 1778 Leonhard Euler determined the explicit formula[1][2]: 12–13
Other equivalent formulas (obtained by expanding this formula) that may be convenient include
The corresponding explicit formulas for s k {\displaystyle s_{k}} and t k {\displaystyle t_{k}} are:[2]: 13
There are recurrence relations for the square triangular numbers, as well as for the sides of the square and triangle involved. We have[3]: (12)
We have[1][2]: 13
All square triangular numbers have the form b 2 c 2 {\displaystyle b^{2}c^{2}} , where b c {\displaystyle {\tfrac {b}{c}}} is a convergent to the continued fraction expansion of 2 {\displaystyle {\sqrt {2}}} , the square root of 2.[4]
A. V. Sylwester gave a short proof that there are infinitely many square triangular numbers: If the n {\displaystyle n} th triangular number n ( n + 1 ) 2 {\displaystyle {\tfrac {n(n+1)}{2}}} is square, then so is the larger 4 n ( n + 1 ) {\displaystyle 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.[5]
The generating function for the square triangular numbers is:[6]
According to the records, it was presented to the St. Petersburg Academy on May 4, 1778.
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