# The Ulam Spiral

## Step (1)

The table below shows
all the triangular numbers multiplied by 2
in the left column
&
the respective triangular numbers
generated from these
in the right column.

## Step (2)

The tabulation below shows
the triangular numbers from
the right column in
the table in step (1)
multiplied by 10
& the resulting product
increased by 1.

## Step (3)

The tabulation below shows
all the sub-totals of the
numbers generated by
the operations described in step (2),
resulting in all whole numbers
to the 5th. power.

##### Triangularnumber

0

(0 x 1)/2 = 0

2

(2 x 3)/2 = 3

6

(6 x 7)/2 = 21

12

(12 x 13)/2 = 78

20

(20 x 21)/2 = 210

## Step (2)

The tabulation below shows
the triangular numbers from
the right column in
the table in step (1)
multiplied by 10
& the resulting product
increased by 1.

10(0) + 1 = 1
10(3) + 1 = 31
10(21) + 1 = 211
10(78) + 1 = 781
10(210) + 1 = 2101

## Step (3)

The tabulation below shows
all the sub-totals of the
numbers generated by
the operations described in step (2),
resulting in all whole numbers
to the 5th. power.

1 = 15
1 + 31 = 32 = 25
1 + 31 + 211 = 243 = 3 5
1 + 31 + 211 + 781 = 1024 = 45
1 + 31 + 211 + 781 + 2101 = 3125= 55

From the tabulation above
one can deduce that in general
n to the 5th. power is
the summation of triangular numbers
per the formula shown below: