A248416 Rectangular array by antidiagonals: for n >= 0, row n gives the positions in the Thue-Morse sequence A010059 at which the first 2^n terms occur.

1, 4, 1, 6, 4, 1, 7, 7, 7, 1, 10, 11, 13, 13, 1, 11, 13, 21, 25, 25, 1, 13, 16, 25, 41, 49, 49, 1, 16, 19, 31, 49, 81, 97, 97, 1, 18, 21, 37, 61, 97, 161, 193, 193, 1, 19, 25, 41, 73, 121, 193, 321, 385, 385, 1, 21, 28, 49, 81, 145, 241, 385, 641, 769, 769, 1, 24, 31, 55, 97, 161, 289, 481, 769, 1281, 1537, 1537, 1

Offset: 1

Author

Clark Kimberling, Oct 06 2014

Keywords

• nonn
• tabl

Comments

Each row contains contains its successor as a proper subsequence.

Note that this supposes that the Thue-Morse sequence A010059 has offset 1, whereas the true offset is 0. So really the entries should all be reduced by 1. - N. J. A. Sloane, Jul 01 2016

Apparently T(n,3) = A004119(n+1) for n>0. Apparently T(n,4) = A083575(n) for n>0. - R. J. Mathar, Nov 06 2018

Examples

			Northwest corner, n>=0, k>=1:
   1    4    6    7   10   11   13   16   18   19
   1    4    7   11   13   16   19   21   25   28
   1    7   13   21   25   31   37   41   49   55
   1   13   25   41   49   61   73   81   97  109
   1   25   49   81   97  121  145  161  193  217
   1   49   97  161  193  241  289  321  385  433
   1   97  193  321  385  481  577  641  769  865
The Thue-Morse sequence A010059 begins with 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, from which we see that the first 4 terms (=1,0,0,1) occur at positions 1, 7, 13, ..., as indicated for row n=2.
		

Crossrefs

Cf. A010059 (Thue-Morse), A026147 (row 0), A091855 (row 1?), A157971 (row 2?),

Column 1 is essentially A004119 (or A181565).

Programs

Maple

A010060 := proc(n)
    local i;
    add(i, i=convert(n, base, 2)) mod 2 ;
end proc:
A010059 := proc(n)
    1-A010060(n) ;
end proc:
A248416Off0 := proc(n,k)
    option remember ;
    local strtN,binpat,src,thue ;
    if k = 1 then
        strtN := 0 ;
    else
        strtN := 1+procname(n,k-1) ;
    end if;
    binpat := [seq(A010059(i),i=0..n-1)] ;
    for src from strtN do
        thue := [seq(A010059(i),i=src..src+nops(binpat)-1)] ;
        if binpat=thue then
            return src ;
        end if;
    end do:
end proc:
A248416 := proc(n,k)
    1+A248416Off0(2^n,k) ;
end proc:
for d from 1 to 11 do
    for k from d to 1 by -1 do
        printf("%d,",A248416(d-k,k)) ;
end do: # R. J. Mathar, Nov 06 2018

Mathematica

z = 3000; u = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {1, 0}}] &, {0}, 20]; Length[u]
t[p_, q_] := t[p, q] = Table[u[[k]], {k, p, q}];
r[n_] := Select[Range[z], t[#, # + 2^(n - 1)] == t[1, 1 + 2^(n - 1)] &]
TableForm[Table[r[n], {n, 0, 10}]]

Extensions

Definitions and examples clarified. - R. J. Mathar, Nov 06 2018