I observed some patterns in some series involving(adsbygoogle = window.adsbygoogle || []).push({});

taking the expression x^n and dividing it by n!, and looking at the remainder for each x, starting with x=0 or x=1

I was wondering if there is some kind of explanation of the patterns that seem to be present

For example, looking at the remainders of the divisions

X^3/3!

or

X^4/6

starting with x=1 and increasing x by 1 will yield (each number separated by a space)

1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 ...

Which shows an obvious pattern

Also, looking at higher n:

X^4/4!

or

X^4 /24

starting with x=1 and increasing x by 1 will yield (each number separated by a space)

1 16 9 16 1 0 1 16 9 16 1 0

which seems to be a repeating string of 1 16 9 16 1 0

with each number being a perfect square between 0 and 24

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Another example, looking at the remainders of

X^5/5!

or

X^5/120

yields

1 32 3 64 5 96 7 8 9 40 11 72 13 104 15 16

There is a pattern here where each odd-positioned number in the series

is simply the value of its place in the list,

whereas the even-positioned numbers always increase by 32 until they exceed 120, and then "overflow" over the value of 120 and so the next value starts over with the overflow added to 0

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Also, there is something more involved I noticed with

x^3/3!

when you look at the QUOTIENT

For the first several values of x, these are the quotients (integer division)

0 1 4 10 20 36 57 85 121 166 221 288 366 457 562 682 818 972 1143 1333

Now I did a process where I take the difference of each two adjacent numbers and create a new series.

(Ex. 1st number in new series is 1-0, 2nd is 4-1, 3rd is 10-4, 4th is 20-10)

This yields

1 3 6 10 16 21 28 36 45 55 67 78 91 105 120 136 154 171 190

Now doing the process AGAIN yields

2 3 4 6 5 7 8 9 10 12 11 13 14 15 16 18 17 19

This is a pattern here where there are 4 numbers that increase steadily by 1, then the next two numbers in the series are transposed (switched) from this normal ordering.

Then there are four more numbers in normal order, then two that are switched, etc.

Note that at the beginning there are only three numbers in order. I think this is because I started generating numbers with x=1 and not x=0

I was wondering if there is are any explanations why these patterns exist, and I would think there are several other patterns involving this idea of dividing x^n/n!

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# Patterns of remainders when dividing x^n/n!

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