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A095190
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Doubled Thue-Morse sequence: a(2n) = A010060(n), a(2n+1) = A010060(n).
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7
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0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1
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OFFSET
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0,1
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COMMENTS
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The A010060 sequence replacing 0 with 0,0 and 1 with 1,1.
Let n = Sum(c(k)*2^k), c(k) = 0,1, be the binary form of n, n = Sum(d(k)*3^k), d(k) = 0,1,2, the ternary form, n = Sum(e(k)*5^k), e(k) = 0,1,2,3,4, the base 5 form. Then a(n) = Sum(c(k)+d(k)) mod 2 = Sum(c(k)+e(k)) mod 2.
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LINKS
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Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003. Apparently unpublished. This is a scanned copy of the version that the author sent to me in 2003. - N. J. A. Sloane, Sep 09 2018. See page 2 for a different construction of this same sequence.
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FORMULA
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a(n) = mod(-1 + Sum_{k=0..n} mod(C(n, 2k), 2), 3). - Paul Barry, Jan 14 2005
a(n) = mod(log_2(Sum_{k=0..n} mod(C(n, 2k),2)),2). - Paul Barry, Jun 12 2006
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EXAMPLE
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The Thue-Morse sequence is: 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 ... so a(n) = 0 0 1 1 1 1 0 0 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 0 0 1 1 1 1 0 0 ...
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MATHEMATICA
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a[n_] := Mod[DigitCount[Floor[n/2], 2, 1], 2]; Array[a, 100, 0] (* Amiram Eldar, Jul 28 2023 *)
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PROG
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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