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 6
& the resulting product
increased by 1.

6(0) + 1 = 1
6(1) + 1 = 7
6(3) + 1 = 19
6(6) + 1 = 37
6(10) + 1 = 61




(3)

The tabulation below shows
all the sub-totals of
the numbers per (2),
resulting in all whole numbers
to the 3rd. power.

1 = 1 = 13
1 + 7 = 8 = 23
1 + 7 + 19 = 27 = 33
1 + 7 + 19 + 37 = 64 = 43
1 + 7 + 19 + 37 + 61 = 125 = 53




From the tabulation above
one can deduce that in general
n to the 3rd. power is
the summation of triangular numbers
per the formula shown below:

equation3rdPower