[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






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