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A248416
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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.
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1
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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
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refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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EXAMPLE
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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.
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MAPLE
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local i;
add(i, i=convert(n, base, 2)) mod 2 ;
end proc:
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:
1+A248416Off0(2^n, k) ;
end proc:
for d from 1 to 11 do
for k from d to 1 by -1 do
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MATHEMATICA
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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}]]
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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Definitions and examples clarified. - R. J. Mathar, Nov 06 2018
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STATUS
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approved
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