Self-building the decimal
expansion of Pi
with a(a(n))
Hello
SeqFans,
this is A000796 (decimal
expansion of Pi):
3,
1, 4, 1, 5, 9, 2, 6, 5, 3, 5, 8, 9, 7, 9, 3, 2, 3, 8, 4, 6, 2, 6, 4, 3, 3, 8,
3, 2, 7, 9, 5, 0, 2, 8, 8, 4, 1, 9, 7, 1, 6, 9, 3, 9, 9, 3, 7, 5, 1, 0, 5, 8,
2, 0, 9, 7, 4, 9, 4, 4, 5, 9, 2, 3, 0, 7, 8, 1, 6, 4, 0, 6, 2, 8, 6, 2, 0, 8,
9, 9, 8, 6, 2, 8, 0, 3, 4, 8, 2, 5, 3, 4, 2, 1, 1, 7, 0, 6, 7, ...
At
every step, the sequence T hereunder produces a new chunk of Pi digits.
When
concatenated and provided with the necessary commas, these chunks reproduce
A000796.
The elegant
(!) definition/formula for T is: « a(a(n)) is the nth
chunk of Pi digits ».
T =
4, 3, 1, 3, 5, 26, 6, 5, 2, 8, 10, 26, 19, 26, 2, 31, 10, 1, 7, 6, 1, 2, 2, 10,
2, 9, 19, 26, 5, 0, 23, 10, 10, 1, 13, 19, 3, 7, 26, 2, 26, 26, 2, 19, 5, 11, 5,...
From
now on, T
is simply produced by A000796 (in short, A) thanks to this “translating
table”:
0 in
A is
0 in T
1 in
A is
3 in T
2 in
A is
9 in T
3 in
A is
2 in T
4 in
A is
1 in T
5 in
A is
5 in T
6 in
A is
7 in T
7 in
A is
19 in T
8 in
A is
10 in T
9 in
A is
26 in T
Example;
in the decimal expansion of Pi, the four yellow integers 5, 1, 0, 5 were encoded
in T
by the three yellow integers 5, 11, 5 (when you insert
5 in the formula, you get 5; when you insert 11, you get ‘10’ -- which needs a
comma between 1 and 0):
A = 3,
1, 4, 1, 5, 9, 2, 6, 5, 3, 5, 8, 9, 7, 9, 3, 2, 3, 8, 4, 6, 2, 6, 4, 3, 3, 8,
3, 2, 7, 9, 5, 0, 2, 8, 8, 4, 1, 9, 7, 1, 6, 9, 3, 9, 9, 3, 7, 5, 1, 0, 5, 8, 2, 0, 9,
7, 4, 9, 4, 4, 5, 9, 2, 3, 0, 7, 8, 1, 6, 4, 0, 6, 2, 8, 6, 2, 0, 8, 9, 9, 8,
6, 2, 8, 0, 3, 4, 8, 2, 5, 3, 4, 2, 1, 1, 7, 0, 6, 7, ...
T =
4, 3, 1, 3, 5, 26, 6, 5, 2, 8, 10, 26, 19, 26, 2, 31, 10, 1, 7, 6, 1, 2, 2, 10,
2, 9, 19, 26, 5, 0, 23, 10, 10, 1, 13, 19, 3, 7, 26, 2, 26, 26, 2, 19, 5, 11, 5,...
From
now on, using the translating table,
we extend T
like this:
A =
(...)5, 1, 0, 5, 8, 2, 0, 9, 7,
4, 9, 4, 4, 5, 9, 2, 3, 0, 7, 8,
1, 6, 4, 0, 6, 2, 8, 6, 2, 0, 8, 9,
...
T =
(...) 5, 11, 5, 10, 9, 0, 26, 19, 1, 26, 1, 1, 5, 26, 9,
2, 0, 19, 10, 3, 7, 1, 0, 7, 9, 10, 7, 9, 0, 10, 26, ...
One
last thing must be said: T invisibly starts with a 0 (zero) -- and this 0 corresponds
to the cases where a(n) = 0. This is forced
by the digits 0 in Pi itself.
To
see how the formula a(a(n)) works, here are the
successive chunks of Pi for n = 1 to
47 (only five yellow chunks of Pi
have more than one digit) :
Pi = 3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5, 8, 9, 7, 9,
3, 2, 3, 8, 4, 6, 2, 6, 4, 3, 3, 8, 3, 2, 7, 9,
5, 0
n = 1 2 3 4 5 6 7 8 9 10 11 12 13 14
15 16 17
18 19 20 21 22 23 24 25 26 27 28
29 30
T =
4, 3, 1, 3, 5, 26, 6, 5, 2, 8, 10, 26, 19, 26, 2, 31, 10, 1, 7, 6, 1, 2,
2, 10, 2, 9, 19, 26, 5, 0,
Pi =
2, 8, 8,
4, 1, 9, 7, 1, 6, 9, 3, 9, 9,
3, 7, 5, 1, 0, 5, 8,
2, 0, 9, ...
n = 31 32 33
34 35 36
37 38 39 40 41 42
43 44 45 46 47 48
49 50 51 ...
T = 23,
10, 10, 1, 13, 19,
3, 7, 26, 2, 26, 26, 2, 19, 5, 11, 5, 10, 9, 0, 26, ...
And
this is how T
must be red to reproduce Pi:
For
n= 1 the 1st chunk of digits of Pi is
to be seen under n= 4 [which is 3]
For
n= 2 the 2nd chunk of digits of Pi is
to be seen under n= 3 [which is 1]
For
n= 3 the 3rd chunk of digits of Pi is
to be seen under n= 1 [which is 4]
For
n= 4 the 4th chunk of digits of Pi is
to be seen under n= 3 [which is 1]
For
n= 5 the 5th chunk of digits of Pi is
to be seen under n= 5 [which is 5]
For
n= 6 the 6th chunk of digits of Pi is
to be seen under n=26 [which is 9]
For
n= 7 the 7th chunk of digits of Pi is
to be seen under n= 6 [which is 26]
For
n= 8 the 8th chunk of digits of Pi is
to be seen under n= 5 [which is 5]
For
n= 9 the 9th chunk of digits of Pi is
to be seen under n= 2 [which is 3]
For
n=10 the 10th chunk of
digits of Pi is to be seen under n= 8
[which is 5]
For
n=11 the 11th chunk of
digits of Pi is to be seen under n=10
[which is 8]
For
n=12 the 12th chunk of
digits of Pi is to be seen under n=26
[which is 9]
For
n=13 the 13th chunk of
digits of Pi is to be seen under n=19
[which is 7]
For
n=14 the 14th chunk of
digits of Pi is to be seen under n=26
[which is 9]
For
n=15 the 15th chunk of
digits of Pi is to be seen under n= 2
[which is 3]
For
n=16 the 16th chunk of
digits of Pi is to be seen under n=31
[which is 23]
For
n=17 the 17th chunk of
digits of Pi is to be seen under n=10
[which is 8]
For
n=18 the 18th chunk of
digits of Pi is to be seen under n= 1
[which is 4]
For
n=19 the 19th chunk of
digits of Pi is to be seen under n= 7
[which is 6]
For
n=20 the 20th chunk of
digits of Pi is to be seen under n= 6
[which is 26]
For
n=21 the 21st chunk of
digits of Pi is to be seen under n= 1
[which is 4]
For
n=22 the 22nd chunk of
digits of Pi is to be seen under n= 2
[which is 3]
For
n=23 the 23rd chunk of
digits of Pi is to be seen under n= 2
[which is 3]
For
n=24 the 24th chunk of
digits of Pi is to be seen under n=10
[which is 8]
For
n=25 the 25th chunk of
digits of Pi is to be seen under n= 2
[which is 3]
For
n=26 the 26th chunk of
digits of Pi is to be seen under n= 9
[which is 2]
For
n=27 the 27th chunk of
digits of Pi is to be seen under n=19
[which is 7]
For
n=28 the 28th chunk of
digits of Pi is to be seen under n=26
[which is 9]
For
n=29 the 29th chunk of
digits of Pi is to be seen under n= 5
[which is 5]
For
n=30 the 30th chunk of
digits of Pi is to be seen under n= 0
[which is 0]
For
n=31 the 31st chunk of
digits of Pi is to be seen under n=23
[which is 2]
For
n=32 the 32nd chunk of
digits of Pi is to be seen under n=10
[which is 8]
For
n=33 the 33rd chunk of
digits of Pi is to be seen under n=10
[which is 8]
For
n=34 the 34th chunk of
digits of Pi is to be seen under n= 1
[which is 4]
For
n=35 the 35th chunk of
digits of Pi is to be seen under n=13
[which is 19]
For
n=36 the 36th chunk of
digits of Pi is to be seen under n=19
[which is 7]
For
n=37 the 37th chunk of
digits of Pi is to be seen under n= 3
[which is 1]
For
n=38 the 38th chunk of
digits of Pi is to be seen under n= 7
[which is 6]
For
n=39 the 39th chunk of
digits of Pi is to be seen under n=26
[which is 9]
For
n=40 the 40th chunk of
digits of Pi is to be seen under n= 2
[which is 3]
For
n=41 the 41st chunk of
digits of Pi is to be seen under n=26
[which is 9]
For
n=42 the 42nd chunk of
digits of Pi is to be seen under n=26
[which is 9]
For
n=43 the 43rd chunk of
digits of Pi is to be seen under n= 2
[which is 3]
For
n=44 the 44th chunk of
digits of Pi is to be seen under n=19
[which is 7]
For
n=45 the 45th chunk of
digits of Pi is to be seen under n= 5
[which is 5]
For
n=46 the 46th chunk of
digits of Pi is to be seen under n=11
[which is 10]
For
n=47 the 47th chunk of
digits of Pi is to be seen under n= 5
[which is 5]
...
this column forms T, and
this Pi (with the necessary commas)
My
challenge in building T was to minimize the quantity of “yellow chunks” (only
five of them), and the size of the “translating table” (the largest figure in
the table is 26). Can this be improved?
Best,
É.