The Ulam Spiral

… and many other spirals¬†
inspired by it …

(1)

The column below shows
all the triangular numbers.

0
1
3
6
10



(2)

The tabulation below shows
the numbers per (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




(3)

The tabulation below shows
the numbers per (2)
multiplied by
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




(4)

The tabulation below shows
all the sub-totals of
the numbers per (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