## 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 , , , , , ... (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 is , where is the th convergent to the Continued Fraction of (Ball and Coxeter 1987, p. 59; Conway and Guy 1996). The first few are

 (1)

The Numerators and Denominators give solutions to the Pell Equation
 (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 denote the th Triangular Number and the th Square Number, then
 (3)

Defining
 (4) (5)

then gives the equation
 (6)

(Conway and Guy 1996). Numbers which are simultaneously Triangular and Square Pyramidal also satisfy the Diophantine Equation
 (7)

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

A general Formula for square triangular numbers is

 (8) (9)

The square triangular numbers also satisfy the Recurrence Relation
 (10) (11)

with , , where . A curious product formula for is given by
 (12)

An amazing Generating Function is
 (13)

(Sloane and Plouffe 1995).

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.