Download The Triangular Sequence The Triangular sequence is: T = 〈0, 0, 1, 3

The Triangular Sequence
The Triangular sequence is:
~T = h0, 0, 1, 3, 6, 10, 15, 21, . . .i
0 1 2 3 4 5 6
6 5 4 3 2 1 0
The Triangular Sequence
The Triangular sequence is quadratic.
Terms in the Triangular sequence lie on the parabola y = x ( x −
1)/2.
10
9
8
7
6
5
4
3
2
1
0
Graph of the Triangular Sequence
1
0 1 2 3 4 5
The Triangular Sequence in Pascal’s Rectangle
The Triangular sequence is column 2 of Pascal’s rectangle.
0
1
2
3
n 4
5
6
7
8
9
0
Binomial Coefficients (nk)
Choose k
1 2 3
4
5
6 7
8
9
1
1
1
1
1
1
1
1
1
1
0
1
2
3
4
5
6
7
8
9
0
0
0
0
0
0
0
0
1
9
0
0
0
0
0
0
0
0
0
1
0
0
1
3
6
10
15
21
28
36
0
0
0
1
4
10
20
35
56
84
0
0
0
0
1
5
15
35
70
126
0
0
0
0
0
1
6
21
56
126
0 0
0 0
0 0
0 0
0 0
0 0
1 0
7 1
28 8
84 36
The Triangular Sequence is the Sum of Gauss Terms
Summing terms in the Gauss sequence produces the Triangular
sequence.
• The sum of no Gauss terms (the empty sum) is 0, term 0 in the
Triangular sequence.
• The sum of one Gauss term, 0 = 0, term 1 in the Triangular
sequence.
• The sum of two Gauss terms, 0 + 1 = 1, term 2 in the Triangular
sequence.
• The sum of three Gauss terms, 0 + 1 + 2 = 3, term 3 in the
Triangular sequence.
• The sum of four Gauss terms, 0 + 1 + 2 + 3 = 6, term 4 in the
Triangular sequence.
2
Computing Terms in the Triangular Sequence
Terms in the Triangular sequence can be computed by the function
n ( n − 1)
t(n) =
(for all natural numbers n)
2
Triangular terms can also be computed by an initial condition
t0 = 0
(the first term is 0)
and a recurrence equation
t n = t n −1 + ( n − 1 )
(the next term equals the previous term plus n − 1), ∀n ∈ N, n ≥ 1
3