A self-building sequence

with the help of Pi

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, ...

And this is the same sequence, replacing the n^th digit above by the a[60+a(n)]^th digit herunder:

5, 9, 1, 9, 2, 3, 4, 10, 2, 5, 2, 8, 3, 7, 3, 5, 4, 5, 8, 1, 10, 4, 10, 1, 5, 5, 8, 5, 4, 7, 3, 2, 6, 4, 8, 8, 1, 9, 3, 7, 9, 10, 3, 5, 3, 3, 5, 7, 2, 9, 6, 2, 8, 4, 6, 3, 7, 1, 3, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 6, 10, 4, 8, 10, 4, 6, 8, 3, 3, 8, 10, 4, 8, 6, 5, 1, 8, 4, 2, 5, 1, 4, 9, 9, 7, 6, 10 ,7 ,3 , ...

The fun (?) part being that you have to use the second sequence to... build the second sequence!

Everything becomes obvious when you consider this presentation (with n clearly visible):

A000796 with n:

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

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,

n = 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

A = 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,

n = 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90

A = 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,

n = 91 92 93 94 95 96 97 98 99 100, ...

A = 5, 3, 4, 2, 1, 1, 7, 0, 6, 7, ...

We see that the first chunk of 10 consecutive different digits in Pi starts at n=61.

(Note that the second such chunk starts at... n=62.)

Giving n’s to the second sequence (named S) shows the substitution (in yellow) -- and explains

the replacement formula a[60+a(n)]:

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

S = 5, 9, 1, 9, 2, 3, 4, 10, 2, 5, 2, 8, 3, 7, 3, 5, 4, 5, 8, 1, 10, 4, 10, 1, 5, 5, 8, 5, 4, 7,

n = 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

S = 3, 2, 6, 4, 8, 8, 1, 9, 3, 7, 9, 10, 3, 5, 3, 3, 5, 7, 2, 9, 6, 2, 8, 4, 6, 3, 7, 1, 3, 1,

n = 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90

S = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 6, 10, 4, 8, 10, 4, 6, 8, 3, 3, 8, 10, 4, 8, 6, 5, 1, 8, 4,

n = 91 92 93 94 95 96 97 98 99 100, ...

S = 2, 5, 1, 4, 9, 9, 7, 6, 10, 7, ...

My two (light) self-cents.

Best,

É.