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,
...