The Ulam Spiral

… and many other spirals¬†
inspired by it …

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.

Triangular Numbers Table
Base
number
Triangular
number
baseNumber
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:

equation5thPower

5th.- power numbers derived from triangular numbers