The Ulam Spiral

… and many other spirals 
inspired by it …

Step (1)

The column below shows
all the triangular numbers.

Step (2)

The tabulation below shows
the numbers from step (1)
multiplied by 4
& the resulting product
increased by 1.

Step (3)

The tabulation below shows
the numbers generated by the
operations described in step (2),
multiplied by the
respective uneven numbers
starting from 1.

Step (4)

The tabulation below shows
all the sub-totals of the
the numbers generated by the
operations described in step (3),
resulting in all whole numbers
to the 4th. power.

0
1
3
6
10




Step (2)

The tabulation below shows
the numbers from step (1)
multiplied by 4
& the resulting product
increased by 1.

4(0) + 1 = 1
4(1) + 1 = 5
4(3) + 1 = 13
4(6) + 1 = 25
4(10) + 1 = 41




Step (3)

The tabulation below shows
the numbers generated by the
operations described in step (2),
multiplied by the
respective uneven numbers
starting from 1.

1 x 1 = 1
3 x 5 = 15
5 x 13 = 65
7 x 25 = 175
9 x 41 = 369




Step (4)

The tabulation below shows
all the sub-totals of the
the numbers generated by the
operations described in step (3),
resulting in all whole numbers
to the 4th. power.

 

1 = 14
1 + 15 = 16 = 24
1 + 15 + 65 = 81 = 34
1 + 15 + 65 + 175 = 256 = 44
1 + 15 + 65 + 175 + 369 = 625 = 54




From the tabulation above
one can deduce that in general
n to the 4th. power is
the summation of triangular numbers
per the formula shown below:

equation4thPower

4th.- power numbers derived from triangular numbers