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A004718
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The Danish composer Per Norgard [Nørgård]'s "infinity sequence", invented in an attempt to unify in a perfect way repetition and variation: a(2n) = -a(n), a(2n+1) = a(n) + 1, a(0)=0.
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4
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0, 1, -1, 2, 1, 0, -2, 3, -1, 2, 0, 1, 2, -1, -3, 4, 1, 0, -2, 3, 0, 1, -1, 2, -2, 3, 1, 0, 3, -2, -4, 5, -1, 2, 0, 1, 2, -1, -3, 4, 0, 1, -1, 2, 1, 0, -2, 3, 2, -1, -3, 4, -1, 2, 0, 1, -3, 4, 2, -1, 4, -3, -5, 6, 1, 0, -2, 3, 0, 1, -1, 2, -2, 3, 1, 0, 3, -2, -4, 5, 0, 1, -1, 2, 1, 0
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OFFSET
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0,4
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COMMENTS
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Minima are at n=2^i-2, maxima at 2^i-1, zeros at A083866.
a(n) has parity of Thue-Morse sequence on {0,1} (A010060).
a(n) = A000120(n) for all n in A060142.
The composer Per Norgard's name is also written in the OEIS as Per Noergaard.
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REFERENCES
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J.-P. Allouche and J. Shallit, The ring of k-regular sequences, II, Theoret. Computer Sci., 307 (2003), 3-29.
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LINKS
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N. J. A. Sloane, First 10000 terms
J.-P. Allouche, J. Shallit, The Ring of k-regular Sequences, II
Per Noergaard [Norgard], Home Page
Per Noergaard [Norgard], "Voyage into the golden screen", 2nd movement
Per Noergaard [Norgard], "Voyage into the golden screen" (MP3 Recording)
Per Noergaard [Norgard], First 128 notes of the infinity series (MP3 Recording)
R. Stephan, Divide-and-conquer generating functions. I. Elementary sequences
Robert Walker, Self Similar Sloth Canon Number Sequences
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FORMULA
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Write n in binary and read from left to write, starting with 0 and interpreting 1 as "add 1" and 0 as "change sign". For example 19 = binary 10011, giving 0 -> 1 -> -1 -> 1 -> 2 -> 3, so a(19) = 3.
G.f.: sum{k>=0, x^(2^k)/[1-x^(2*2^k)] * prod{l=0, k-1, x^(2^l)-1}}.
The g.f. satisfies F(x^2)(1-x) = F(x)-x/(1-x^2).
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MAPLE
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f:=proc(n) option remember; if n=0 then RETURN(0); fi; if n mod 2 = 0 then RETURN(-f(n/2)); else RETURN(f((n-1)/2)+1); fi; end;
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MATHEMATICA
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a[n_?EvenQ] := a[n]= -a[n/2]; a[0]=0; a[n_] := a[n]= a[(n-1)/2]+1; Table[a[n], {n, 0, 85}](* From Jean-François Alcover, Nov 18 2011 *)
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PROG
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(PARI) a=vector(100); a[1]=1; a[2]=-1; for(n=3, #a, a[n]=if(n%2, a[n\2]+1, -a[n\2])); a \\ Charles R Greathouse IV, Nov 18 2011
(Haskell)
a004718 n = foldr ($) 0 $ noergaard n where
noergaard x | x == 0 = []
| d == 0 = negate : noergaard x'
| d == 1 = (+ 1) : noergaard x' where (x', d) = divMod x 2
-- Reinhard Zumkeller, Nov 10 2012
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CROSSREFS
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Sequence in context: A145579 A167655 A157218 * A157225 A055347 A055288
Adjacent sequences: A004715 A004716 A004717 * A004719 A004720 A004721
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KEYWORD
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sign,nice,easy,changed
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AUTHOR
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Jorn B. Olsson (olsson(AT)math.ku.dk)
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EXTENSIONS
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Edited by Ralf Stephan, Mar 07 2003
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STATUS
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approved
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