Hello Seq-Fans,

The sequence S(1) was inspired to me by a clever remark from Alexandre Wajnberg:

S(1) = 0,3,11,4,5,21,32,26,...

If you concatenate, for i=1 to inf., all [a(i)+a(i+1)], you’ll get the decimal expansion of Pi (A000796).

Let’s see:

a(1)=0

a(2)=3

and a(1)+a(2)=3

a(2)=3

a(3)=11

and a(2)+a(3)=14

a(3)=11

a(4)=4

and a(3)+a(4)=15

a(4)=4

a(5)=5

and a(4)+a(5)=9

etc.

The same method, applied to S(2), S(3) and S(4), leads to the same result:

S(2) = 1,2,12,3,6,20,33,25, ...

S(3) = 2,1,0,4,11,81,572,17, ...

S(4) = 3,0,1,3,12,80,573,16, ...

Note that S(5) produces the same result -- and so do (infinitely) more sequences of this type:

S(5) = 0,31,10,49,16,19,70, ...

S(6) is monotonic:

S(6) = 0,3,11,148,2505, ...

I’ve not decided how to deal with the zeros in Pi’s decimal expansion yet. Any idea?

Is there any of those sequences worth entering the OEIS?

What about the expansions of e, sqr2, phi, etc.?

The rule could be extended to the sum of three consecutive terms:

S(7) = 0,1,2,11,2,79,572, ...

Best,

É.

A000796 = 3,1,4,1,5,9,2,6,5,3,5,8,9,7,9,3,2,3,8,4,6,2,6,4,3,3,8,3,2,7,9,5,0,2,8,8,4,1,9,7,1,6,9,3,9,9,3,7,5,1,0,5,8,2,0,9,7,4,9,4,4,5,9,2,3,0,7,8,1,6,4,0,6,2,8,6,2,0,8,9,9,8,6,2,8,0,3,4,8,2,5,3,4,2,1,1,7,0,6,7,9,8,2,1,4, ...