a binomial classification for all integers

[wouter meeussen]

classify the integers according to the count of carries in the binary addition of n with n+2.
Example:
n=13, n+2=15, binary {1,1,0,1}.{1,1,1,1}=3           (*dot product counts paired 1's*)
n=35, n+2=37, binary {1,0,0,0,1,1}.{1,0,0,1,0,1}=2

Now increment the carry count with 1, and
for all integers upto 2^4-1, this classification produces
.    1    2    3    4  = column_index

.     1    2    6   14
      3    4   10  
      7    5   12
    15    8   13
           9
          11
where the ' .' stands for the missing zero, completing the binomial table.

First column is 2^n-1 (A000225)
second column: A099628
third and further columns: nihil
Some similarity to A087136: Smallest number m such that A073642(m)=n.
First row: 2^n-2 except for initial 1 (A095121)
Second row: nihil.

I withold submission temporarily until I figured out a convenient formatting,
and got some time to explore this further. Eventual suggestions & bug-alerts are welcome, of course.

Wouter.

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Mathematica 4.0:

w[n_Integer]:={First[#][[1]],Last[#]}& /@ Transpose/@Split[Sort[{(IntegerDigits[#,2,n].IntegerDigits[#+2,2,n]),#}& /@ Range[2^n/2,2^n-1]],First[#1]===First[#2]&]

with a colourfull presentation of them:
TableForm[ Transpose[w2[7]] /. k_Integer:> StyleForm[c=(Floor[Log[2.,k+1]])/8.;k+1,FontColor->Hue[c]],TableAlignments->{Left,Top},TableSpacing->{0,1}]
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