# [seqfan] A075681 and A003878

israel at math.ubc.ca israel at math.ubc.ca
Thu Sep 10 01:10:39 CEST 2015

```What to do about these sequences?

A003878's Name is n^4+(9/2)*n^3+n^2-(9/2)*n+1. The original Name was
"Number of stacks of n pikelets, distance 4 flips from a well-ordered
stack". Now the formula in the Name would imply a(0) to a(4) are 1, 3, 48,
199, 543, which are the first five numbers in Data, but the Offset is given
as 3 which would make these a(3) to a(7). Is it safe to assume that Offset
is just a mistake, and should be 0? The given g.f. also fits with an Offset
of 0, rather than 3. I don't know about the "pikelet" interpretation
though.

A075681 is "Difference between (n-1)*(n-2)^3 and A003878." It is given as
having offset 1 and Data starting 0, 0, 2, 21, 60, 121, 207, 321, and the
Formula from Ralf Stephan is 1/2*n^3+7/2*n^2-23n+25 for n > 2. Well, Ralf's
formula gives 6, -3, 1, 21, 60, 121, 207 for n = 1 .. 7. Moreover, the
difference between (n-1)*(n-2)^3 and Ralf's formula is -6, 3, 1, 3, 48,
199, 543 for n = 1 .. 7. So, what do we make of this? The Data would fit
with "Difference between (n-1)*(n-2)^3 and A003878" for n >= 4 if A003878
keeps its current (presumably wrong) offset of 3. Should A075681(3) be 1
instead of 2? Should we define A075681 in terms of A003878(n+3) if the
offset of A003878 is changed to 0? Should we discard the two 0's and give
A075681 an offset of 3? Does the current situation make sense in terms of
the "flips" cryptically referred to in the Example?

Cheers,
Robert

```