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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. 18
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.

Comment from Michael Nyvang on the "iris" score on the "Voyage into the golden screen" video, Dec 31 2018: That is A004718 on the cover in the 12-tone tempered chromatic scale. The music - as far as I recall - is constructed from this base by choosing subsequences out of this sequence in what Per calls 'wave lengths', and choosing different scales modulo (to-tone, overtones on one fundamental, etc). There quite a lot more to say about this, but I believe this is the foundation. - N. J. A. Sloane, Jan 05 2019

From Antti Karttunen, Mar 09 2019: (Start)

This sequence can be represented as a binary tree. After a(0) = 0 and a(1) = 1, each child to the left is obtained by negating the parent node's contents, and each child to the right is obtained by adding one to the parent's contents:

                                      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

etc.

Sequences A323907, A323908 and A323909 are in bijective correspondence with this sequence and their terms are all nonnegative.

(End)

LINKS

N. J. A. Sloane, First 10000 terms

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

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

Yu Hin Au, Christopher Drexler-Lemire & Jeffrey Shallit, Notes and note pairs in Nørgård's infinity series, Journal of Mathematics and Music, Volume 11, 2017, Issue 1, pages 1-19. - N. J. A. Sloane, Dec 31 2018

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

Per Nørgård [Noergaard], The infinity series, on YouTube

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

Per Nørgård [Noergaard], Voyage into the golden screen, on YouTube.

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

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

Jon Wild, Comments on the musical score in the YouTube illustrations for the "Iris" and "Voyage into the golden screen" videos

Index entries for sequences related to music

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

Zumkeller's formula implies that a(2n) = -a(n), and so a(n) = a(4n) = a(16n) = .... - N. J. A. Sloane, Dec 31 2018

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, A323886, A323887, A323907, A323908, A323909.

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 October 20 12:47 EDT 2019. Contains 328257 sequences. (Running on oeis4.)