:

The first 7 partial repeating decimal sequences of sin(10^-n) for integer n

Maple

I define a partial repeating decimal as shown in the following example: if you have the decimal expansion 0.1728394877777777777777777777771939374652819101093837... 7 is called a partial repeating decimal.

Back in 2000 I noticed a pattern in the decimal expansions of sin(10^-n) for growing n. Here is table of some integer n:

n                sin(10^-n)

1 9.98334166*10^-2

2 9.99983333416666468*10^-3

3 9.9999983333334166666646825*10^-4

4 9.9999999833333333416666666646825396*10^-5

5 9.99999999983333333333416666666666468253968254*10^-6

6 9.999999999998333333333333416666666666664682539682539100*10^-7

7 9.9999999999999833333333333333416666666666666646825396825396828152*10^-8

8 9.99999999999999983333333333333333416666666666666666468253968253968254243827*10^-9

We see the first five partial repeating decimals are {0,9,3,6, 682539} For some very large n there appears to be two more decimal sequences, repeating a few times, that only occur with certain given n's. For example the sixth repeating decimal sequence is listed below.

n=      11, 14, 17, 20, 23, 26, 29, ... 200 and some others give 382716049.

n=      12, 15, 18, 21, 24, 27, 30, ...300 and some others give  097001763668430335.

n=10, 13, 16, 19, 22, 25, 28, 31, ...100 and some others give  525573192239858906.

For the 7th partial repeating sequence in sin(10^-n) we have the following:

766394099727433060 appears in n=14, 17, 20, 23, 26,29,...,50, 53, 56,...,99,101,104,and some others.

798140131473464806 appears in n=10,13,15,16,18,19,21,22,24,..., 51,52,54,55,57,...,100,102,103,and some others.

﻿