minor numiracle

wouter meeussen wouter.meeussen at pandora.be
Fri May 2 18:40:03 CEST 2003

```Superseeker can sometimes be miraculous:

I fed it a sequence 1,2,6,20,76,296,1240,5200,22960,100512,458592

generated by counting the length of the string
{0} ; {-1,1} ; {-2,0,2, -2,0,2};
or "replace k with {-1-Abs[k], ... , 1+Abs[k]} with increment 2".

and it replied with:
[-2n^2+n^3+(4n+5n^2) a[n] + (-2-6n-6n^2-2n^3) a[n]^2, revogf]

this juggles into
CoefficientList[InverseSeries[
Series[(-4n-5n^2+n^2Sqrt[1+8n+8n^2])/(2(-2-6n-6n^2-2n^3)),{n,0,12}] ],n]

How does LISTTOALGEQ work this? Who wrote that?

same sequence, starting from {1} gives
{1} ; {-2,0,2} ; {-3,-1,1,3 , -1,1 , -3,-1,1,3}
with lengths counted by
(2, 6, 20, 76, 296, 1240, 5200,...) /2

if you count the sums of the Abs instead of lengths,
you get, starting from {1}:
1, 4, 18, 72, 324, 1360, 6280
which is just
{3, 10, 38, 148, 620, 2600} - {2, 6, 20, 76, 296, 1240}

and starting from {0}:
0, 2, 8, 36, 144, 648, 2720
which is twice the previous shifted right.

************************
Counting lengths of nested lists generates familiar seq's,
as
A000108 from k -> 0,..,k+1
A001003 from k-> 0,..,k+1,..,0
A006319 from k -> -Abs[k],..,Abs[k]+1
with again a link to (inverse binomial transform of A071356).

************************

Wouter.

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