A085144 -id:A085144 - cp's OEIS frontend

A085145 Positions of 0 in A085144.

0, 1, 3, 7, 10, 15, 21, 26, 31, 36, 43, 46, 53, 58, 63, 73, 82, 87, 93, 100, 107, 110, 117, 122, 127, 136, 147, 150, 156, 165, 170, 175, 180, 187, 190, 201, 210, 215, 221, 228, 235, 238, 245, 250, 255, 273, 290, 295, 301, 313, 324, 331, 334, 341

Offset: 1

Ralf Stephan, Jun 20 2003

If n is odd then (n-1)/2 is also in sequence, with the even subsequence starting 0,10,26,36,46,58...

Records for first differences are 2,3,...,2^k+2,2^k+3...

A255723 Another variant of Per Nørgård's "infinity sequence", cf. A004718: t(0) = 0; t(4n) = t(n); t(4n+1) = t(n) - 2; t(4n+2) = -t(n) - 1; t(4n+3) = t(n) + 2.

Original entry on oeis.org

0, -2, -1, 2, -2, -4, 1, 0, -1, -3, 0, 1, 2, 0, -3, 4, -2, -4, 1, 0, -4, -6, 3, -2, 1, -1, -2, 3, 0, -2, -1, 2, -1, -3, 0, 1, -3, -5, 2, -1, 0, -2, -1, 2, 1, -1, -2, 3, 2, 0, -3, 4, 0, -2, -1, 2, -3, -5, 2, -1, 4, 2, -5, 6, -2, -4, 1, 0, -4, -6, 3, -2, 1, -1

Offset: 0

Reinhard Zumkeller, Mar 19 2015

sign

Per Nørgård's surname is also written as Noergaard;

example of a sequence sharing with A004718 some main characterizing properties, see link (chapter 7).

Yu Hin (Gary) Au, Christopher Drexler-Lemire, and Jeffrey Shallit, "Notes and note pairs in Norgard's infinity series", J. of Mathematics and Music (2017). DOI: http://dx.doi.org/10.1080/17459737.2017.1299807

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Christopher Drexler-Lemire, Jeffrey Shallit, Notes and Note-Pairs in Noergaard's Infinity Series, arXiv:1402.3091 [math.CO], page 12f.
Jørgen Mortensen, "Is Per Nørgård's infinity series unique?"

Cf. A004718, A256184, A087808, A085144.

a255723 n = a255723_list !! n
a255723_list = 0 : concat (transpose [map (subtract 2) a255723_list,
                                      map (-1 -) a255723_list,
                                      map (+ 2) a255723_list,
                                      tail a255723_list])

A256184 First of two variations by Per Nørgård of his "infinity sequence", cf. A004718: u(0) = 0; u(3n) = -u(n); u(3n+1) = u(n) - 2; u(3*n+2) = u(n) - 1.

Original entry on oeis.org

0, -2, -1, 2, -4, -3, 1, -3, -2, -2, 0, 1, 4, -6, -5, 3, -5, -4, -1, -1, 0, 3, -5, -4, 2, -4, -3, 2, -4, -3, 0, -2, -1, -1, -1, 0, -4, 2, 3, 6, -8, -7, 5, -7, -6, -3, 1, 2, 5, -7, -6, 4, -6, -5, 1, -3, -2, 1, -3, -2, 0, -2, -1, -3, 1, 2, 5, -7, -6, 4, -6, -5

Offset: 0

Reinhard Zumkeller, Mar 19 2015

sign

Per Nørgård's surname is also written as Noergaard.

Not squarefree in contrast to A004718, first repetition of order 3: a(32) = a(33) = a(34) = -1, see link.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Yu Hin (Gary) Au, Christopher Drexler-Lemire, and Jeffrey Shallit, Notes and note pairs in Norgard's infinity series, J. of Mathematics and Music (2017).
Christopher Drexler-Lemire and Jeffrey Shallit, Notes and Note-Pairs in Noergaard's Infinity Series, arXiv:1402.3091 [math.CO], 2014, page 13.

Cf. A004718, A256185, A087808, A085144, A255723.

a256184 n = a256184_list !! n
a256184_list = 0 : concat (transpose [map (subtract 2) a256184_list,
                                      map (subtract 1) a256184_list,
                                      map negate $ tail a256184_list])

from functools import lru_cache
@lru_cache(maxsize=None)
def a(n): return 0 if n == 0 else (a(n//3) - (3-n%3)) if n%3 else -a(n//3)
print([a(n) for n in range(72)]) # Michael S. Branicky, Sep 02 2021

A256185 Second of two variations by Per Nørgård of his "infinity sequence", cf. A004718: v(0) = 0; v(3n) = -v(n); v(3n+1) = v(n) - 3; v(3*n+2) = -2 - v(n).

Original entry on oeis.org

0, -3, -2, 3, -6, 1, 2, -5, 0, -3, 0, -5, 6, -9, 4, -1, -2, -3, -2, -1, -4, 5, -8, 3, 0, -3, -2, 3, -6, 1, 0, -3, -2, 5, -8, 3, -6, 3, -8, 9, -12, 7, -4, 1, -6, 1, -4, -1, 2, -5, 0, 3, -6, 1, 2, -5, 0, 1, -4, -1, 4, -7, 2, -5, 2, -7, 8, -11, 6, -3, 0, -5, 0

Offset: 0

Reinhard Zumkeller, Mar 19 2015

sign

Per Nørgård's surname is also written as Noergaard;

for all odd j exists k such that abs(a(k+1)-a(k)) = j, in contrast to A004718, where this holds also for even j > 0, see link.

Yu Hin (Gary) Au, Christopher Drexler-Lemire, and Jeffrey Shallit, "Notes and note pairs in Norgard's infinity series", J. of Mathematics and Music (2017). DOI: http://dx.doi.org/10.1080/17459737.2017.1299807

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Christopher Drexler-Lemire, Jeffrey Shallit, Notes and Note-Pairs in Noergaard's Infinity Series, arXiv:1402.3091 [math.CO], page 13

Cf. A004718, A256184, A087808, A085144, A255723.

a256185 n = a256185_list !! n
a256185_list = 0 : concat (transpose [map (subtract 3) a256185_list,
                                      map (-2 -) a256185_list,
                                      map negate $ tail a256185_list])

cp's OEIS Frontend

A085145 Positions of 0 in A085144.

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A255723 Another variant of Per Nørgård's "infinity sequence", cf. A004718: t(0) = 0; t(4n) = t(n); t(4n+1) = t(n) - 2; t(4n+2) = -t(n) - 1; t(4n+3) = t(n) + 2.

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Haskell

A256184 First of two variations by Per Nørgård of his "infinity sequence", cf. A004718: u(0) = 0; u(3n) = -u(n); u(3n+1) = u(n) - 2; u(3*n+2) = u(n) - 1.

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Programs

Haskell

Python

A256185 Second of two variations by Per Nørgård of his "infinity sequence", cf. A004718: v(0) = 0; v(3n) = -v(n); v(3n+1) = v(n) - 3; v(3*n+2) = -2 - v(n).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

cp's OEIS Frontend

A085145 Positions of 0 in A085144.

Views

Author

Keywords

Comments

A255723 Another variant of Per Nørgård's "infinity sequence", cf. A004718: t(0) = 0; t(4*n) = t(n); t(4*n+1) = t(n) - 2; t(4*n+2) = -t(n) - 1; t(4*n+3) = t(n) + 2.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

A256184 First of two variations by Per Nørgård of his "infinity sequence", cf. A004718: u(0) = 0; u(3*n) = -u(n); u(3*n+1) = u(n) - 2; u(3*n+2) = u(n) - 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Python

A256185 Second of two variations by Per Nørgård of his "infinity sequence", cf. A004718: v(0) = 0; v(3*n) = -v(n); v(3*n+1) = v(n) - 3; v(3*n+2) = -2 - v(n).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

A255723 Another variant of Per Nørgård's "infinity sequence", cf. A004718: t(0) = 0; t(4n) = t(n); t(4n+1) = t(n) - 2; t(4n+2) = -t(n) - 1; t(4n+3) = t(n) + 2.

A256184 First of two variations by Per Nørgård of his "infinity sequence", cf. A004718: u(0) = 0; u(3n) = -u(n); u(3n+1) = u(n) - 2; u(3*n+2) = u(n) - 1.

A256185 Second of two variations by Per Nørgård of his "infinity sequence", cf. A004718: v(0) = 0; v(3n) = -v(n); v(3n+1) = v(n) - 3; v(3*n+2) = -2 - v(n).