Mathematical_Recreations-Kraitchik-2e

Numerical Pastimes 69
1 (n + d - 2)! . .
Nd,i = d! (n _ I)! [m - (t - d)].
The sum of the first n terms of this flequence gives the nth
figurate number of dimension (d + 1) and difference i:
When i =
Nd+l,i = "LNd,i'
'i= 1
1, we have
1 (n + d - I)!
Nd,l = d! en _ 1)! = (n+d-l)Cd = (n+d-l)Cn - b
that is, N d.l is the number
of combinations of n + d - 1
things taken d at a time or
n - 1 at a time.
When i = d we have
1 (n + d - 2)!
Nd,d = (d _ 1) ( (n _ I)! ·n.
Thus
Nl,l = n,
N 2 ,2 = n2,
FIGURE 9. Geometrical In-
Na,a = tn2(n + 1), terpretation " h
0 t e Figurate
N 4 ,4 = tn2(n + l)(n + 2), Number of Diophantus.
Ns,s = nn2(n + 1)(n + 2)(n + 3),
Ndd = 1 n2(n + 1) ... (n + d - 2).
, (d - I)!
When d is larger than 1, Nd,d is exactly divisible by n 2 •
Here is a formula due to Diophantus:
8T+ 1 = S,
where T and S are corresponding triangular and square numbers.
The geometrical interpretation of this is given in Figure
9. Algebraically it means,
8· n(n 2+ 1) + 1 = (2n + 1)2.
n