The Ulam Spiral

… and many other spirals 
inspired by it …

Step (1)

The table below shows
all the whole numbers
in the left column
&
the respective triangular numbers
generated from these
in the right column.

Step (2)

The tabulation below shows
the addition of each pair of
adjacent triangular numbers from
the right column in
the table in step (1),
resulting in all whole numbers
to the 2nd. power.

Triangular Numbers Table
Base
number
Triangular
number
baseNumber
triangularNumber

0

(0 x 1)/2 = 0

1

(1 x2 )/2 = 1

2

(2 x 3)/2 = 3

3

(3 x 4)/2 = 6

4

(4 x 5)/2 = 10

5

(5 x 6)/2 = 15

Step (2)

The tabulation below shows
the addition of each pair of
adjacent triangular numbers from
the right column in
the table in step (1),
resulting in all whole numbers
to the 2nd. power.

(0 x 1)/2 + (1 x 2)/2   = 0 + 1 = 1 = 12

(1 x 2)/2 + (2 x 3)/2   = 1 + 3 = 4 = 22

(2 x 3)/2 + (3 x 4)/2   = 3 + 6 = 9 = 32

(3 x 4)/2 + (4 x 5)/2   = 6 + 10 = 16 =  42
(4 x 5)/2 + (5 x 6)/2   = 10 + 15 = 25 =  52




From the tabulation above
one can deduce that in general
in to the 2nd. power is
the sum of 2 triangular numbers
per the formula shown below:

equation2ndPower

2nd.- power numbers derived from triangular numbers