When I started working on the formula for square pyramidal numbers I had
already found the formula for triangular numbers with a little help from my
brother and the formula for triangular pyramidal numbers. The reason
discovering the answer to this is of note is that while the pattern in the
other two series was apparent fairly early, the pattern for square pyramidal
numbers was not.
A square pyramidal number (spn) is the sum of all whole number squares
up to and including n. For example if
n=4 than spn= 30 or 1+4+9+16. I started working on the problem by
making a table showing n, n2,
and spn.
n
|
n2
|
spn
|
1
|
1
|
1
|
2
|
4
|
5
|
3
|
9
|
14
|
4
|
16
|
30
|
5
|
25
|
55
|
6
|
36
|
91
|
7
|
49
|
140
|
8
|
64
|
204
|
9
|
81
|
285
|
10
|
100
|
385
|
Next, I started looking for a pattern by looking at the relationship
between n and spn. Which gave me this table:
spn
|
n
|
spn/n
|
1
|
1
|
1
|
5
|
2
|
2 ½
|
14
|
3
|
4 ⅔
|
30
|
4
|
7 ½
|
55
|
5
|
11
|
91
|
6
|
15 ⅙
|
140
|
7
|
20
|
204
|
8
|
25 ½
|
285
|
9
|
31 ⅔
|
385
|
10
|
38 ½
|
Examining the fractions I see that to get whole number answers in each of
the relationships, I need to multiply by 6.
spn
|
6spn
|
n
|
6spn/n
|
1
|
6
|
1
|
6
|
5
|
30
|
2
|
15
|
14
|
84
|
3
|
28
|
30
|
180
|
4
|
45
|
55
|
330
|
5
|
66
|
91
|
546
|
6
|
91
|
140
|
840
|
7
|
120
|
204
|
1224
|
8
|
153
|
285
|
1710
|
9
|
190
|
385
|
2310
|
10
|
231
|
Now I could start to see a pattern. In each case, 6spn/n is divisible by n+1. 6sp2/2 is
evenly divisible by 3. 6sp3/3 is
evenly divisible by 4.
spn
|
6spn
|
n
|
6spn/n
|
(6spn/n)/(n+1)
|
1
|
6
|
1
|
6
|
3
|
5
|
30
|
2
|
15
|
5
|
14
|
84
|
3
|
28
|
7
|
30
|
180
|
4
|
45
|
9
|
55
|
330
|
5
|
66
|
11
|
91
|
546
|
6
|
91
|
13
|
140
|
840
|
7
|
120
|
15
|
204
|
1224
|
8
|
153
|
17
|
285
|
1710
|
9
|
190
|
19
|
385
|
2310
|
10
|
231
|
21
|
Now the pattern is fully evident. The results are counting odds starting
at 3. More importantly, the new results are n+(n+1). So, 6spn is
n(n+1)(n+n+1) or the pyramidal number for n is the product of n, the next whole
number (n+1) and the sum of n and the next whole number divided by six. Written
out fully the formula is:
n(n+1)(2n+1)
6
I call this a discovery not because I was the first
person to find this formula. Wikipedia has several pages about triangular and
pyramidal numbers. I call this a discovery because I went in search for it and
found it. I was not taught about pyramidal numbers. I never looked it up until
I started writing this piece. The discovery was finding it.
Too often we discount the idea of discovery by
requiring a person to be the first. I think this breaks the spirit of
discovery, damages born curiosity. We should all be searching and trying to
discover. Finding things out for ourselves. I have figured out triangular
numbers and two forms of pyramidal numbers for myself. I have discovered them,
though they were solved by other people before me. I went on to figure out a
formula for the addition of cubed numbers. I am thinking hexagonal numbers will
be next, then hexagonal pyramids.
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