[seqfan] Re: Golay-Rudin-Shapiro sequences

Kevin Ryde user42 at zip.com.au
Tue Jun 12 04:06:15 CEST 2012

njasloane at gmail.com (Neil Sloane) writes:
> I have been cleaning up a bunch of sequences related to the
> Golay-Rudin-Shapiro (aka Rudin-Shapiro) sequence (or word).
> The main sequence is A020985.

I was tinkering recently with the connection of those to the alternate
paper folding curve A106665.  I think that may be well-known, but my
cribs as follows if it might inspire cross references.

    A020985 -- Golay/Rudin/Shapiro sequence
                 dX and dY, skipping every second value zero
                 dSum, change in X+Y

                 dX = GRS(N) if N even
                      0      if N odd

                 dY = 0      if N even
                      GRS(N) if N odd

                 dSum = dX + dY = GRS(N)

    A020986 -- Golay/Rudin/Shapiro cumulative
                 X coordinate undoubled

    A020990 -- Golay/Rudin/Shapiro * (-1)^n, cumulative
                 Y coordinate undoubled
                 X-Y diff, starting from N=1

I take the curve as first step along the X axis to X=1,Y=0 and then
upwards to X=1,Y=1, etc.  By "undoubled" I mean the X coordinate doubles
up as 0 then 1,1,2,2,etc and A020986 is every second such value.  Is the
jargon "bisection"?  I suppose it's X=A020986(ceil(n/2)) or something
like that, starting from n=0.

Bit more notes I made under "dX,dY" code at
but I worry it's not reading very clearly yet :-)

Even the white bits were black.

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