 |  |  |
(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 
, and are the sum of consecutive triangular
numbers. The first few are 1, 4, 10, 20, 35, 56, 84, 120, ... (OEIS A000292).
The generating function for the tetrahedral
numbers is
 |
(4)
|
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 
,
,
and 
(giving 
,
,
and 
),
as proved by Meyl (1878; cited in Dickson 2005, p. 25).
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; Avanesov 1966/1967; Mordell 1969, p. 258; 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 
.
The minimum number of tetrahedral numbers needed to sum to 
, 2, ... are given by 1, 2, 3, 1, 2, 3, 4, 2, 3, 1, ... (OEIS
A104246). Pollock's
conjecture states that every number is the sum of at most five tetrahedral numbers.
The numbers that are not the sum of 
tetrahedral numbers are given by the sequence 17, 27,
33, 52, 73, ..., (OEIS A000797) containing
241 terms, with 
being almost certainly the last such number.
