Square Triangular Number

A number which is simultaneously Square and Triangular. The first few are 1, 36, 1225, 41616, 1413721, 48024900, ... (Sloane's A001110), corresponding to $T_1=S_1$, $T_8=S_6$, $T_{49}=S_{35}$, $T_{288}=S_{204}$, $T_{1681}=S_{1189}$, ... (Pietenpol 1962), but there are an infinite number, as first shown by Euler in 1730 (Dickson 1952).

The general Formula for a square triangular number ${\it ST}_n$ is $b^2c^2$, where $b/c$ is the $n$th convergent to the Continued Fraction of $\sqrt{2}$ (Ball and Coxeter 1987, p. 59; Conway and Guy 1996). The first few are

$${1\over 1}, {3\over 2}, {7\over 5}, {17\over 12}, {41\over 29}, {99\over 70}, {239\over 169}, \cdots.$$ (1)

The Numerators and Denominators give solutions to the Pell Equation

$$x^2-2y^2=\pm 1,$$ (2)

but can also be obtained by doubling the previous Fraction and adding to the Fraction before that. The connection with the Pell Equation can be seen by letting $N$ denote the $N$th Triangular Number and $M$ the $M$th Square Number, then

$${\textstyle{1\over 2}}N(N+1)=M^2.$$ (3)

Defining

$$x \equiv 2N+1$$ (4)

$$y \equiv 2M$$ (5)

then gives the equation

$$x^2-2y^2=1$$ (6)

(Conway and Guy 1996). Numbers which are simultaneously Triangular and Square Pyramidal also satisfy the Diophantine Equation

$$3(2y+1)^2=8x^3+12x^2+4x+3.$$ (7)

The only solutions are $x=-1$, 0, 1, 5, 6, and 85 (Guy 1994, p. 147).

A general Formula for square triangular numbers is

$${\it ST}_n = \left[{(1+\sqrt{2}\,)^{2n}-(1-\sqrt{2}\,)^{2n}\over 4\sqrt{2}}\right]^2$$ (8)

$$= {\textstyle{1\over 32}} [(17+12\sqrt{2})^n+(17-12\sqrt{2})^n-2].$$ (9)

The square triangular numbers also satisfy the Recurrence Relation

$${\it ST}_n = 34{\it ST}_{n-1}-{\it ST}_{n-2}+2$$ (10)

$$u_{n+2} = 6u_{n+1}-u_n,$$ (11)

with $u_0=0$, $u_1=1$, where ${\it ST}_n\equiv{u_n}^2$. A curious product formula for ${\it ST}_n$ is given by

$${\it ST}_n=2^{2n-5}\prod_{k=1}^{2n}\left[{3+\cos\left({k\pi\over n}\right)}\right].$$ (12)

An amazing Generating Function is

$$f(x)={1+x\over(1-x)(1-34x+x^2)}=1+36x+1225x^2+\ldots$$ (13)

(Sloane and Plouffe 1995).

See also Square Number, Square Root, Triangular Number

References

Allen, B. M. ``Squares as Triangular Numbers.'' Scripta Math. 20, 213-214, 1954.

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, 1987.

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 203-205, 1996.

Dickson, L. E. A History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Chelsea, pp. 10, 16, and 27, 1952.

Guy, R. K. ``Sums of Squares'' and ``Figurate Numbers.'' §C20 and §D3 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 136-138 and 147-150, 1994.

Khatri, M. N. ``Triangular Numbers Which are Also Squares.'' Math. Student 27, 55-56, 1959.

Pietenpol, J. L. ``Square Triangular Numbers.'' Problem E 1473. Amer. Math. Monthly 69, 168-169, 1962.

Sierpinski, W. Teoria Liczb, 3rd ed. Warsaw, Poland: Monografie Matematyczne t. 19, p. 517, 1950.

Sierpinski, W. ``Sur les nombres triangulaires carrés.'' Pub. Faculté d'Électrotechnique l'Université Belgrade, No. 65, 1-4, 1961.

Sierpinski, W. ``Sur les nombres triangulaires carrés.'' Bull. Soc. Royale Sciences Liège, 30 ann., 189-194, 1961.

Sloane, N. J. A. Sequence A001110/M5259 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.

Walker, G. W. ``Triangular Squares.'' Problem E 954. Amer. Math. Monthly 58, 568, 1951.