The Ulam Spiral

… and many other spirals¬†
inspired by it …

(1)

The table below shows
all the triangular numbers multiplied by 2
as the base numbers &
the respective triangular numbers
generated from these.

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

(2)

The tabulation below shows
the numbers per (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




(3)

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