The 
th
tetrahedral number 
is a figurate number defined for a positive
integer 
by
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
where 
is the 
th
triangular number and 
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 
.
The first few are 1, 4, 10, 20, 35, 56, 84, 120, ... (OEIS A000292,
with the initial 0 omitted). The generating function
for the tetrahedral numbers is
 |
(4)
|
for 
.
The tetrahedral number 
is odd precisely when 
and is even
otherwise (Conway and Guy 1996).
The only positive integers which are simultaneously square and tetrahedral are 
, 
, and 
(giving 
, 
, and 
), as proved by Meyl (1878; cited in Dickson 2005,
p. 25).
For positive integers 
and 
, numbers which are simultaneously triangular
and tetrahedral satisfy the binomial coefficient
equation
 |
(5)
|
the only solutions of which are
 |  |  |
(6)
|
 |  |  |
(7)
|
 |  |  |
(8)
|
 |  |  |
(9)
|
 |  |  |
(10)
|
(OEIS A027568, with the initial 0 omitted; Avanesov 1967, pp. 409-410; Mordell 1969, p. 258; Guy 1994, p. 147).
Beukers and Top (1988, p. 204) studied the problem of finding numbers which are simultaneously tetrahedral and square pyramidal
using integer points on an
elliptic curve. They proved that the only positive
solution is 
.
The minimum numbers of tetrahedral summands needed to represent 
, 2, ... are given by 1, 2, 3, 1, 2, 3, 4, 2, 3, 1, ... (OEIS
A104246). Pollock's
conjecture states that every positive integer
is the sum of at most five tetrahedral numbers. The known
numbers that are not the sum of at most four tetrahedral
numbers begin 17, 27, 33, 52, 73, ... (OEIS A000797).
There are 241 known terms, the largest being 343867. It is conjectured that there
are no further terms.
