Triangular Number

A Figurate Number of the form $T_n\equiv n(n+1)/2$ obtained by building up regular triangles out of dots. The first few triangle numbers are 1, 3, 6, 10, 15, 21, ... (Sloane's A000217). $T_4=10$ gives the number and arrangement of Bowling pins, while $T_5=15$ gives the number and arrangement of balls in Billiards. Triangular numbers satisfy the Recurrence Relation

$${T_{n+1}}^2-{T_n}^2=(n+1)^3,$$ (1)

as well as

$$3T_n+T_{n-1} = T_{2n}$$ (2)

$$3T_n+T_{n+1} = T_{2n+1}$$ (3)

$$1+3+5+\ldots+(2n-1) = T_n+T_{n-1}$$ (4)

and

$$(2n+1)^2=8T+1=T_{n-1}+6T_n+T_{n+1}$$ (5)

(Conway and Guy 1996). They have the simple Generating Function

$$f(x)={x\over (1-x)^3}=x+3x^2+6x^3+10x^4+15x^5+\ldots.$$ (6)

Every triangular number is also a Hexagonal Number, since

$${\textstyle{1\over 2}}r(r+1)=\cases{ \left({r+1\over 2}\rig... ...eft[{2\left({-{r\over 2}}\right)-1}\right]& for $r$\ even.\cr}$$ (7)

Also, every Pentagonal Number is 1/3 of a triangular number. The sum of consecutive triangular numbers is a Square Number, since

$$T_r+T_{r-1} = {\textstyle{1\over 2}}r(r+1)+{\textstyle{1\over 2}}(r-1)r$$

$$= {\textstyle{1\over 2}}r[(r+1)+(r-1)] = r^2.$$ (8)

Interesting identities involving triangular numbers and Square Numbers are

$$\sum_{k=1}^{2n-1} (-1)^{k+1}T_k=n^2$$ (9)

$${T_n}^2=\sum_{k=1}^n k^3 ={\textstyle{1\over 4}}n^2(n+1)^2$$ (10)

$$\sum_{k=1, 3, \ldots, q} k^3=T_n$$ (11)

for $q$ Odd and

$$n={\textstyle{1\over 2}}(q^2+2q-1).$$ (12)

All Even Perfect Numbers are triangular $T_p$ with Prime $p$. Furthermore, every Even Perfect Number $P>6$ is of the form

$$P=1+9T_n=T_{3n+1},$$ (13)

where $T_n$ is a triangular number with $n=8j+2$ (Eaton 1995, 1996). Therefore, the nested expression

$$9(9\cdots(9(9(9(9T_n+1)+1)+1)+1)\ldots+1)+1$$ (14)

generates triangular numbers for any $T_n$. An Integer $k$ is a triangular number Iff $8k+1$ is a Square Number $>1$.

The numbers 1, 36, 1225, 41616, 1413721, 48024900, ... (Sloane's A001110) are Square Triangular Numbers, i.e., numbers which are simultaneously triangular and Square (Pietenpol 1962). Numbers which are simultaneously triangular and Tetrahedral satisfy the Binomial Coefficient equation

$$T_n={n+1\choose 2}={m+2\choose 3}={\it Te}_m$$ (15)

and are given by ${\it Te}_3=T_4=10$, ${\it Te}_8=T_{15}=120$, ${\it Te}_{20}=T_{55}=1540$, and ${\it Te}_{34}=T_{119}=7140$ (Guy 1994, p. 147).

The smallest of two Integers for which $n^3-13$ is four times a triangular number is 5 (Cesaro 1886; Le Lionnais 1983, p. 56). The only Fibonacci Numbers which are triangular are 1, 3, 21, and 55 (Ming 1989), and the only Pell Number which is triangular is 1 (McDaniel 1996). The Beast Number 666 is triangular, since

$$T_{6\cdot 6}=T_{36}=666.$$ (16)

In fact, it is the largest Repdigit triangular number (Bellew and Weger 1975-76).

Fermat's Polygonal Number Theorem states that every Positive Integer is a sum of most three Triangular Numbers, four Square Numbers, five Pentagonal Numbers, and $n$ $n$-Polygonal Numbers. Gauß proved the triangular case, and noted the event in his diary on July 10, 1796, with the notation

$$** E\Upsilon RHKA\qquad {\it num}=\Delta+\Delta+\Delta.$$ (17)

This case is equivalent to the statement that every number of the form $8m+3$ is a sum of three Odd Squares (Duke 1997). Dirichlet derived the number of ways in which an Integer $m$ can be expressed as the sum of three triangular numbers (Duke 1997). The result is particularly simple for a Prime of the form $8m+3$, in which case it is the number of squares mod $8m+3$ minus the number of nonsquares mod $8m+3$ in the Interval $4m+1$ (Deligne 1973).

The only triangular numbers which are the Product of three consecutive Integers are 6, 120, 210, 990, 185136, 258474216 (Guy 1994, p. 148).

See also Figurate Number, Pronic Number, Square Triangular Number

References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, p. 59, 1987.

Bellew, D. W. and Weger, R. C. ``Repdigit Triangular Numbers.'' J. Recr. Math. 8, 96-97, 1975-76.

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 33-38, 1996.

Deligne, P. ``La Conjecture de Weil.'' Inst. Hautes Études Sci. Pub. Math. 43, 273-308, 1973.

Dudeney, H. E. Amusements in Mathematics. New York: Dover, pp. 67 and 167, 1970.

Duke, W. ``Some Old Problems and New Results about Quadratic Forms.'' Not. Amer. Math. Soc. 44, 190-196, 1997.

Eaton, C. F. ``Problem 1482.'' Math. Mag. 68, 307, 1995.

Eaton, C. F. ``Perfect Number in Terms of Triangular Numbers.'' Solution to Problem 1482. Math. Mag. 69, 308-309, 1996.

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.

Hindin, H. ``Stars, Hexes, Triangular Numbers and Pythagorean Triples.'' J. Recr. Math. 16, 191-193, 1983-1984.

Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 56, 1983.

McDaniel, W. L. ``Triangular Numbers in the Pell Sequence.'' Fib. Quart. 34, 105-107, 1996.

Ming, L. ``On Triangular Fibonacci Numbers.'' Fib. Quart. 27, 98-108, 1989.

Pappas, T. ``Triangular, Square & Pentagonal Numbers.'' The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 214, 1989.

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

Satyanarayana, U. V. ``On the Representation of Numbers as the Sum of Triangular Numbers.'' Math. Gaz. 45, 40-43, 1961.

Sloane, N. J. A. Sequences A000217/M2535 and 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.