Tetrahedral Number

A Figurate Number ${\it Te}_n$ of the form

$${\it Te}_n=\sum_{i=1}^n T_n = {\textstyle{1\over 6}} n(n+1)(n+2)={n+2\choose 3},$$ (1)

where $T_n$ is the $n$th Triangular Number and ${n\choose m}$ is a Binomial Coefficient. These numbers correspond to placing discrete points in the configuration of a Tetrahedron (triangular base pyramid). Tetrahedral numbers are Pyramidal Numbers with $r=3$, and are the sum of consecutive Triangular Numbers. The first few are 1, 4, 10, 20, 35, 56, 84, 120, ... (Sloane's A000292). The Generating Function of the tetrahedral numbers is

$${x\over(x-1)^4}=x+4x^2+10x^3+20x^4+\ldots.$$ (2)

Tetrahedral numbers are Even, except for every fourth tetrahedral number, which is Odd (Conway and Guy 1996).

The only numbers which are simultaneously Square and Tetrahedral are ${\it Te}_1=1$, ${\it Te}_2=4$, and ${\it Te}_{48}=19600$ (giving $S_1=1$, $S_2=4$, and $S_{140}=19600$), as proved by Meyl (1878; cited in Dickson 1952, p. 25). Numbers which are simultaneously Triangular and tetrahedral satisfy the Binomial Coefficient equation

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

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). Beukers (1988) has studied the problem of finding numbers which are simultaneously tetrahedral and Pyramidal via Integer points on an Elliptic Curve, and finds that the only solution is the trivial ${\it Te}_1=P_1=1$.

See also Pyramidal Number, Truncated Tetrahedral Number

References

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

Beukers, F. ``On Oranges and Integral Points on Certain Plane Cubic Curves.'' Nieuw Arch. Wisk. 6, 203-210, 1988.

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 44-46, 1996.

Dickson, L. E. History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Chelsea, 1952.

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

Meyl, A.-J.-J. ``Solution de Question 1194.'' Nouv. Ann. Math. 17, 464-467, 1878.

Sloane, N. J. A. Sequence A000292/M3382 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.