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A004718 The Danish composer Per 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. 10
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

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 Nørgård's name is also written in the OEIS as Per Noergaard.

REFERENCES

J.-P. Allouche and J. Shallit, The ring of k-regular sequences, II, Theoret. Computer Sci., 307 (2003), 3-29.

LINKS

N. J. A. Sloane, First 10000 terms

J.-P. Allouche, J. Shallit, The Ring of k-regular Sequences, II

Christopher Drexler-Lemire, Jeffrey Shallit, Notes and Note-Pairs in Noergaard's Infinity Series, arXiv:1402.3091 [math.CO], 2014.

Per Nørgård [Noergaard], Home Page

Per Nørgård [Noergaard], "Voyage into the golden screen", 2nd movement

Per Nørgård [Noergaard], "Voyage into the golden screen" (MP3 Recording)

Per Nørgård [Noergaard], First 128 notes of the infinity series (MP3 Recording)

R. Stephan, Divide-and-conquer generating functions. I. Elementary sequences, arXiv:math/0307027 [math.CO], 2003.

Robert Walker, Self Similar Sloth Canon Number Sequences

Wikipedia, Unendlichkeitsreihe

FORMULA

Write n in binary and read from left to right, 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).

a(n) = (2 * (n mod 2) - 1) * a(floor(n/2)) + n mod 2. - Reinhard Zumkeller, Mar 20 2015

MAPLE

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;

MATHEMATICA

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}](* Jean-François Alcover, Nov 18 2011 *)

Table[Fold[If[#2 == 0, -#1, #1 + 1] &, IntegerDigits[n, 2]], {n, 0, 85}] (* Michael De Vlieger, Jun 30 2016 *)

PROG

(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)

import Data.List (transpose)

a004718 n = a004718_list !! n

a004718_list = 0 : concat

   (transpose [map (+ 1) a004718_list, map negate $ tail a004718_list])

-- Reinhard Zumkeller, Mar 19 2015, Nov 10 2012

CROSSREFS

Cf. A083866, A256187 (first differences); variants: A256184, A256185, A255723.

Sequence in context: A167655 A262781 A157218 * A157225 A055347 A055288

Adjacent sequences:  A004715 A004716 A004717 * A004719 A004720 A004721

KEYWORD

sign,nice,easy,hear

AUTHOR

Jorn B. Olsson (olsson(AT)math.ku.dk)

EXTENSIONS

Edited by Ralf Stephan, Mar 07 2003

STATUS

approved

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Last modified December 9 12:25 EST 2016. Contains 278971 sequences.