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Talk:More Transformations of Integer Sequences

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Would some Administrator please edit the protected page More Transformations of Integer Sequences and remove {{Edit request}} here.

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

LPAL needs the special case LPAL1 = IDENTITY. It could alternatively be written as
B(x) = (A(x)+A(x^2)) / (1-A(x^2))

CHKk is listed as
bn = (MÖBIUS · AIK)kan / n
but it should be
bn = (MÖBIUS · AIK)kan / k

CIKk can be given directly using Pólya enumeration on the cyclic group as
bn = sum{d|k and d|n} phi(d) (AIKk/da)n/d / k
This not only shows how to skip CHK in calculating CIK but also makes explicit the calculation of CIK2, which is referred to by CPAL2. Otherwise we're left to derive it independently or guess that if CIK = MÖBIUS-1 · CHK then maybe CIK2 = MÖBIUS-1 · CHK2.

Which brings me on to CPAL. The component J of the case n,k is given as
J=sum{i=1 to n/2}(AIKk/2-1a)(n-2i)/2
which accounts for only n-2i items and k-2 boxes. It should be
J=sum{i=1 to n/2} ai (AIKk/2-1a)(n-2i)/2

The explanation could also be clearer: what exactly is the joining operation of "boxes joined"?

However, working backwards from DIKk = (CIKk + CPALk) / 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:

If n odd, k even:
bn = sum{i>0 and i<n/2} (AIK2a)n-2i × (AIK(k-2)/2a)i / 2
If n,k even:
bn = (AIKk/2a)n/2 / 2 + sum{i>0 and i<n/2} (AIK2a)n-2i × (AIK(k-2)/2a)i / 2
If k odd: bn = sum{i>0 and i<n/2} an-2i × (AIK(k-1)/2a)i

Or even simpler:

2B(x) = (1+A(x))2 / (1 - A(x2)) - 1

It might also be nice to include LPAL and CPAL in the catalogue.

LPAL sequences

A057427 LPAL s1
A060546 LPAL s2
A056449 LPAL s3
A056450 LPAL s4
A056451 LPAL s5
A163403 LPAL all1
A068911 (LPAL all2) + s1n+1
A003945 (LPAL all3) + s1n+1
A000045 (LPAL codd)2n + s1n+1
A000045 (LPAL codd)2n-3
A000045 (LPAL noone)2n-2 + s1n
A000045 (LPAL noone)2n+1
A000045 (LPAL twoone)n-3 + s1n+1

CPAL sequences

A057427 CPAL s1
A029744 CPAL s2
A182751 CPAL s3
A056486 CPAL s4
A052955 (CPAL all1)n+1
A068156 (CPAL all3) + s1n+1
A000071 (CPAL codd)2n-4 - s1n-1
A000045 (CPAL codd)2n-3
A000071 (CPAL noone)2n-4 - s1n-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)