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A005536
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a(0) = 0; thereafter a(2n) = n - a(n) - a(n-1), a(2n+1) = n - 2a(n) + 1.
(Formerly M2274)
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6
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0, 1, 0, 0, 1, 3, 3, 4, 3, 3, 1, 0, 0, 1, 0, 0, 1, 3, 3, 4, 6, 9, 10, 12, 12, 13, 12, 12, 13, 15, 15, 16, 15, 15, 13, 12, 12, 13, 12, 12, 10, 9, 6, 4, 3, 3, 1, 0, 0, 1, 0, 0, 1, 3, 3, 4, 3, 3, 1, 0, 0, 1, 0, 0, 1, 3, 3, 4, 6, 9, 10, 12, 12, 13, 12, 12, 13, 15, 15, 16, 18, 21, 22, 24, 27, 31, 33
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OFFSET
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0,6
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COMMENTS
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A "Von Koch" sequence generated by the first Feigenbaum symbolic sequence.
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. G. Stanton, W. L. Kocay and P. H. Dirksen, Computation of a combinatorial function, pp. 569-578 of C. J. Nash-Williams and J. Sheehan, editors, Proceedings of the Fifth British Combinatorial Conference 1975. Utilitas Math., Winnipeg, 1976.
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LINKS
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FORMULA
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Partial sums of A065359. a(n) = Sum_{i=0..n} Sum_{k=0..i} (-1)^k*(floor(i/2^k) - 2*floor(i/2^(k+1))). - Benoit Cloitre, Mar 28 2004
G.f.: (1/(1-x)^2) * Sum_{k>=0} (-1)^k*x^2^k/(1 + x^2^k). - Ralf Stephan, Apr 27 2003
a(n) = -n*(n-2) + 3*Sum_{k=1..n-1} Sum_{i=1..k} A035263(i+1), where A035263 is the first Feigenbaum symbolic sequence. - Benoit Cloitre, May 29 2003
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MATHEMATICA
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a[n_] := a[n] = If[n == 0, 0, hn = Floor[n/2]; If[OddQ[n], hn - 2 a[hn] + 1, hn - a[hn] - a[hn - 1]]]; t = Table[a[n], {n, 0, 100}] (* T. D. Noe, Mar 22 2012 *)
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PROG
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(PARI) a(n)=-n*(n-2)+3*sum(k=1, n-1, sum(i=1, k, abs(subst(Pol(binary(i+1))- Pol(binary(i)), x, 1)%2))) \\ Benoit Cloitre, May 29 2003
(PARI) a(n)=polcoeff(1/(1-x)^2*sum(k=0, 10, (-1)^k*x^2^k/(1+x^2^k)) +O(x^(n+1)), n)
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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More terms and better description from Ralf Stephan, Sep 13 2003
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STATUS
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approved
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