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A277238
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Folding numbers (see comments for the definition).
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4
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1, 2, 6, 10, 12, 22, 28, 38, 42, 52, 56, 78, 90, 108, 120, 142, 150, 170, 178, 204, 212, 232, 240, 286, 310, 346, 370, 412, 436, 472, 496, 542, 558, 598, 614, 666, 682, 722, 738, 796, 812, 852, 868, 920, 936, 976, 992, 1086, 1134, 1206, 1254, 1338, 1386, 1458, 1506, 1596, 1644, 1716, 1764, 1848, 1896, 1968
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OFFSET
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1,2
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COMMENTS
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Folding numbers: Numbers with an even number of bits in their binary expansion such that the XOR of the left half and the reverse of the right half is the all-1's string. Numbers with an odd number of bits in their binary expansion such that the central bit is 1, and the XOR of the left (n-1)/2 bits and the reverse of the right (n-1)/2 bits is the all-1's string.
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LINKS
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EXAMPLE
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178 in base 2 is 10110010. Taking the XOR of 1011 and 0100 (which is 0010 reversed) gives the result 1111, so 178 is in the sequence.
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MAPLE
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N:= 16: # to get all terms < 2^N
M[1]:= [[1]]: M[2]:= [[1, 0]]:
for d from 3 to N by 2 do
M[d]:= map(L -> [op(L[1..(d-1)/2]), 1, op(L[(d+1)/2..-1])], M[d-1]);
if d < N then
M[d+1]:= map(L -> ([op(L[1..(d-1)/2]), 0, 1, op(L[(d+1)/2..-1])], [op(L[1..(d-1)/2]), 1, 0, op(L[(d+1)/2..-1])]), M[d-1])
fi
od:
seq(seq(add(L[-i]*2^(i-1), i=1..d), L=M[d]), d=1..N); # Robert Israel, Nov 09 2016
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MATHEMATICA
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{1}~Join~Select[Range@ 2000, If[OddQ@ Length@ # && Take[#, {Ceiling[ Length[#]/2]}] == {0}, False, Union[Take[#, Floor[Length[#]/2]] + Reverse@ Take[#, -Floor[Length[#]/2]]] == {1}] &@ IntegerDigits[#, 2] &] (* Michael De Vlieger, Oct 07 2016 *)
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PROG
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(PARI) isok(n) = {if (n==1, return(1)); b = binary(n); if ((#b % 2) && (b[#b\2+1] == 0), return (0)); vecmin(vector(#b1, k, bitxor(b[k], b[#b-k+1]))) == 1; } \\ Michel Marcus, Oct 07 2016
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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