Square Triangular Number
A number which is simultaneously square and triangular. Let 
denote the 
th triangular
number and 
the 
th square
number, then a number which is both triangular and square satisfies the equation
, or
 |
(1)
|
Completing the square gives
Therefore, defining
gives the Pell equation
 |
(8)
|
(Conway and Guy 1996). The first few solutions are 
, (17,
12), (99, 70), (577, 408), .... These give the solutions 
, (8,
6), (49, 35), (288, 204), ... (OEIS A001108
and A001109), corresponding to the triangular
square numbers 1, 36, 1225, 41616, 1413721, 48024900, ... (OEIS A001110;
Pietenpol 1962). In 1730, Euler showed that there are an infinite number of such
solutions (Dickson 2005).
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
 |
(9)
|
(OEIS A001333 and A000129). The numerators and denominators
can also be obtained by doubling the previous fraction
and adding to the fraction before that.
A general formula for square triangular numbers is
The square triangular numbers also satisfy the recurrence
relation
 |
(12)
|
A second-order recurrence for 
is given
by
 |
(13)
|
with 
and 
. A first-order
recurrence equation is given by
 |
(14)
|
(M. Carreira, pers. comm., Sept. 29, 2003).
A curious product formula for 
is given by
![ST_n=2^(2n-5)product_(k=1)^(2n)[3+cos((kpi)/n)].](/images/equations/SquareTriangularNumber/NumberedEquation7.gif) |
(15)
|
An amazing generating function is
 |
(16)
|
(Sloane and Plouffe 1995).
Taking the square and triangular numbers together gives the sequence 1, 1, 3, 4, 6, 9, 10, 15, 16, 21, 25, ... (OEIS A005214;
Hofstadter 1996, p. 15).
SEE ALSO: Cubic Triangular Number,
Pentagonal Square Number,
Pentagonal
Square Triangular Number,
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. History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Dover,
pp. 10, 16, and 27, 2005.
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.
Hofstadter, D. R. Fluid Concepts & Creative Analogies: Computer Models of the Fundamental Mechanisms
of Thought. New York: Basic Books, 1996.
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.
Potter, D. C. D. "Triangular Square Numbers." Math. Gaz. 56,
109-110, 1972.
Sengupta, D. "Digits in Triangular Squares." College Math. J. 30,
31, 1999.
Sierpiński, W. Teoria Liczb, 3rd ed. Warsaw, Poland: Monografie Matematyczne
t. 19, p. 517, 1950.
Sierpiński, W. "Sur les nombres triangulaires carrés." Pub. Faculté d'Électrotechnique l'Université Belgrade, No. 65,
1-4, 1961.
Sierpiński, W. "Sur les nombres triangulaires carrés." Bull.
Soc. Royale Sciences Liège, 30 ann., 189-194, 1961.
Silverman, J. H. A Friendly Introduction to Number Theory. Englewood Cliffs, NJ: Prentice Hall,
1996.
Sloane, N. J. A. Sequences A000129/M1413, A001333/M2665, A001108/M4536,
A001109/M4217, and A001110/M5259
in "The On-Line Encyclopedia of Integer Sequences."
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.
Referenced on Wolfram|Alpha:
Square Triangular Number
CITE THIS AS:
Weisstein, Eric W. "Square Triangular Number."
From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SquareTriangularNumber.html
![[((1+sqrt(2))^(2n)-(1-sqrt(2))^(2n))/(4sqrt(2))]^2](/images/equations/SquareTriangularNumber/Inline33.gif)
![1/(32)[(17+12sqrt(2))^n+(17-12sqrt(2))^n-2].](/images/equations/SquareTriangularNumber/Inline36.gif)
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