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Talk:More Transformations of Integer Sequences
Links to sequences are broken. — Daniel Forgues 01:42, 10 February 2013 (UTC)
Errata and suggestions
These apply both to the wiki page and to http://oeis.org/transform2.html
LPAL needs the special case LPAL_{1} = IDENTITY. It could alternatively be written as
B(x) = (A(x)+A(x^2)) / (1-A(x^2))
CHK_{k} is listed as
b_{n} = (MÖBIUS · AIK)_{k}a_{n} / n
but it should be
b_{n} = (MÖBIUS · AIK)_{k}a_{n} / k
CIK_{k} can be given directly using Pólya enumeration on the cyclic group as
b_{n} = sum{d|k and d|n} phi(d) (AIK_{k/d}a)_{n/d} / k
This not only shows how to skip CHK in calculating CIK but also makes explicit the calculation of CIK_{2}, which is referred to by CPAL_{2}.
Otherwise we're left to derive it independently or guess that if CIK = MÖBIUS^{-1} · CHK then maybe CIK_{2} = MÖBIUS^{-1} · CHK_{2}.
Which brings me on to CPAL. The component J of the case n,k is given as
J=sum{i=1 to n/2}(AIK_{k/2-1}a)_{(n-2i)/2}
which accounts for only n-2i items and k-2 boxes. It should be
J=sum{i=1 to n/2} a_{i} (AIK_{k/2-1}a)_{(n-2i)/2}
The explanation could also be clearer: what exactly is the joining operation of "boxes joined"?
However, working backwards from DIK_{k} = (CIK_{k} + CPAL_{k}) / 2 and comparing the well-known cycle indices of the cyclic group and the dihedral group, it seems that a simpler rephrasing could be given:
CPAL_{k}
If n odd, k even:
b_{n} = sum{i>0 and i<n/2} (AIK_{2}a)_{n-2i} × (AIK_{(k-2)/2}a)_{i} / 2
If n,k even:
b_{n} = (AIK_{k/2}a)_{n/2} / 2 + sum{i>0 and i<n/2} (AIK_{2}a)_{n-2i} × (AIK_{(k-2)/2}a)_{i} / 2
If k odd:
b_{n} = sum{i>0 and i<n/2} a_{n-2i} × (AIK_{(k-1)/2}a)_{i}
Or even simpler:
CPAL
2B(x) = (1+A(x))^{2} / (1 - A(x^{2})) - 1
It might also be nice to include LPAL and CPAL in the catalogue.
LPAL sequences
A057427 | LPAL s^{1} |
A060546 | LPAL s^{2} |
A056449 | LPAL s^{3} |
A056450 | LPAL s^{4} |
A056451 | LPAL s^{5} |
A163403 | LPAL all^{1} |
A068911 | (LPAL all^{2}) + s^{1}_{n+1} |
A003945 | (LPAL all^{3}) + s^{1}_{n+1} |
A000045 | (LPAL codd)_{2n} + s^{1}_{n+1} |
A000045 | (LPAL codd)_{2n-3} |
A000045 | (LPAL noone)_{2n-2} + s^{1}_{n} |
A000045 | (LPAL noone)_{2n+1} |
A000045 | (LPAL twoone)_{n-3} + s^{1}_{n+1} |
CPAL sequences
A057427 | CPAL s^{1} |
A029744 | CPAL s^{2} |
A182751 | CPAL s^{3} |
A056486 | CPAL s^{4} |
A052955 | (CPAL all^{1})_{n+1} |
A068156 | (CPAL all^{3}) + s^{1}_{n+1} |
A000071 | (CPAL codd)_{2n-4} - s^{1}_{n-1} |
A000045 | (CPAL codd)_{2n-3} |
A000071 | (CPAL noone)_{2n-4} - s^{1}_{n-1} |
A000071 | (CPAL noone)_{2n-1} |
A000071 | (CPAL twoone)_{n-3} |
Independent checking of the offsets is welcomed.
Peter J. Taylor 21:24, 22 September 2017 (UTC)