A124841 -id:A124841 - cp's OEIS frontend

A001030 Fixed under 1 -> 21, 2 -> 211.

2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2

Offset: 1

N. J. A. Sloane

If treated as the terms of a continued fraction, it converges to approximately
2.57737020881617828717350576260723346479894963737498275232531856357441\
7024804797827856956758619431996. - Peter Bertok (peter(AT)bertok.com), Nov 27 2001
There are a(n) 1's between successive 2's. - Eric Angelini, Aug 19 2008
Same sequence where 1's and 2's are exchanged: A001468. - Eric Angelini, Aug 19 2008

Midhat J. Gazale, Number: From Ahmes to Cantor, Section on 'Cleavages' in Chapter 6, Princeton University Press, Princeton, NJ 2000, pp. 203-211.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=1..8119
N. G. de Bruijn, Sequences of zeros and ones generated by special production rules, Indag. Math., 43 (1981), 27-37.
D. R. Hofstadter, Eta-Lore [Cached copy, with permission]
D. R. Hofstadter, Pi-Mu Sequences [Cached copy, with permission]
D. R. Hofstadter and N. J. A. Sloane, Correspondence, 1977 and 1991
A. Nagel, A self-defining infinite sequence, with an application to Markoff chains and probability, Math. Mag., 36 (1963), 179-183.
N. J. A. Sloane, Handwritten notes on Self-Generating Sequences, 1970 (note that A1148 has now become A005282).
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence).

Length of the sequence after 'n' substitution steps is given by the terms of A000129.
Equals A004641(n) + 1.
The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A000201 as the parent: A000201, A001030, A001468, A001950, A003622, A003842, A003849, A004641, A005614, A014675, A022342, A088462, A096270, A114986, A124841. - N. J. A. Sloane, Mar 11 2021

Following Spage's PARI program.
a001030 n = a001030_list !! (n-1)
a001030_list = [2, 1, 1, 2] ++ f [2] [2, 1, 1, 2] where
   f us vs = ws ++ f vs (vs ++ ws) where
             ws = 1 : us ++ 1 : vs
-- Reinhard Zumkeller, Aug 04 2014

('n' is the number of substitution steps to perform.) Nest[Flatten[ # /. {1 -> {2, 1}, 2 -> {2, 1, 1}}] &, {1}, n]
SubstitutionSystem[{1->{2,1},2->{2,1,1}},{2},{6}][[1]] (* Harvey P. Dale, Feb 15 2022 *)

/* Fast string concatenation method giving e.g. 5740 terms in 8 iterations */
a="2";b="2,1,1,2";print1(b);for(x=1,8,c=concat([",1,",a,",1,",b]);print1(c);a=b;b=concat(b,c)) \\ K. Spage, Oct 08 2009

from math import isqrt
def A001030(n): return [2, 1, 1, 2, 1, 2, 1, 2][n-1] if n < 9 else -isqrt(m:=(n-9)*(n-9)<<1)+isqrt(m+(n-9<<2)+2) # Chai Wah Wu, Aug 25 2022

a(n) = -1 + floor(n*(1+sqrt(2))+1/sqrt(2))-floor((n-1)*(1+sqrt(2))+1/sqrt(2)). - Benoit Cloitre, Jun 26 2004. [I don't know if this is a theorem or a conjecture. - N. J. A. Sloane, May 14 2008]

This is a theorem, following from Hofstadter's Generalized Fundamental Theorem of eta-sequences on page 10 of Eta-Lore. See also de Bruijn's paper from 1981 (hint from Benoit Cloitre). - Michel Dekking, Jan 22 2017

More terms from Peter Bertok (peter(AT)bertok.com), Nov 27 2001

A004641 Fixed under 0 -> 10, 1 -> 100.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1

Offset: 1

N. J. A. Sloane

Partial sums: A088462. - Reinhard Zumkeller, Dec 05 2009

Write w(n) = a(n) for n >= 1. Each w(n) is generated by w(i) for exactly one i <= n; let g(n) = i. Each w(i) generates a single 1, in a word (10 or 100) that starts with 1. Therefore, g(n) is the number of 1s among w(1), ..., w(n), so that g = A088462. That is, this sequence is generated by its partial sums. - Clark Kimberling, May 25 2011

T. D. Noe, Table of n, a(n) for n = 1..8119
Wieb Bosma, Michel Dekking, and Wolfgang Steiner, A remarkable sequence related to Pi and sqrt(2), arXiv:1710.01498 [math.NT], 2017.
Wieb Bosma, Michel Dekking, and Wolfgang Steiner, A remarkable sequence related to Pi and sqrt(2), Integers, Electronic Journal of Combinatorial Number Theory 18A (2018), #A4.
N. G. de Bruijn, Sequences of zeros and ones generated by special production rules, Nederl. Akad. Wetensch. Indag. Math. 43 (1981), no. 1, 27-37. Reprinted in Physics of Quasicrystals, ed. P. J. Steinhardt et al., p. 664.
C. J. Glasby, S. P. Glasby, and F. Pleijel, Worms by number, Proc. Roy. Soc. B, Proc. Biol. Sci. 275 (1647) (2008) 2071-2076.
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence).
Index entries for characteristic functions

Equals A001030 - 1. Essentially the same as A006337 - 1 and A159684.
Characteristic function of A086377.
Cf. A081477.
The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A000201 as the parent: A000201, A001030, A001468, A001950, A003622, A003842, A003849, A004641, A005614, A014675, A022342, A088462, A096270, A114986, A124841. - N. J. A. Sloane, Mar 11 2021

[Floor(n*(Sqrt(2) - 1) + Sqrt(1/2)) - Floor((n - 1)*(Sqrt(2) - 1) + Sqrt(1/2)): n in [0..100]]; // Vincenzo Librandi, Mar 27 2015

P(0):= (1,0): P(1):= (1,0,0):
((P~)@@6)([1]);
# in Maple 12 or earlier, comment the above line and uncomment the following:
# (curry(map,P)@@6)([1]); # Robert Israel, Mar 26 2015

Nest[ Flatten[# /. {0 -> {1, 0}, 1 -> {1, 0, 0}}] &, {1}, 5] (* Robert G. Wilson v, May 25 2011 *)
SubstitutionSystem[{0->{1,0},1->{1,0,0}},{1},5]//Flatten (* Harvey P. Dale, Nov 20 2021 *)

from math import isqrt
def A004641(n): return [1, 0, 0, 1, 0, 1, 0, 1][n-1] if n < 9 else -1-isqrt(m:=(n-9)*(n-9)<<1)+isqrt(m+(n-9<<2)+2) # Chai Wah Wu, Aug 25 2022

a(n) = floor(n*(sqrt(2) - 1) + sqrt(1/2)) - floor((n - 1)*(sqrt(2) - 1) + sqrt(1/2)) (from the de Bruijn reference). - Peter J. Taylor, Mar 26 2015
From Jianing Song, Jan 02 2019: (Start)
a(n) = A001030(n) - 1.
a(n) = A006337(n-9) - 1 = A159684(n-10) for n >= 10. (End)

A088462 a(1)=1, a(n) = ceiling((n - a(a(n-1)))/2).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 6, 6, 6, 7, 7, 8, 8, 8, 9, 9, 10, 10, 11, 11, 11, 12, 12, 13, 13, 13, 14, 14, 15, 15, 16, 16, 16, 17, 17, 18, 18, 18, 19, 19, 20, 20, 21, 21, 21, 22, 22, 23, 23, 23, 24, 24, 25, 25, 25, 26, 26, 27, 27, 28, 28, 28, 29, 29, 30, 30, 30, 31, 31, 32, 32

Offset: 1

Benoit Cloitre, Nov 12 2003

nonn

Partial sums of A004641. - Reinhard Zumkeller, Dec 05 2009

This sequence generates A004641; see comment at A004641. - Clark Kimberling, May 25 2011

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)

Cf. A005206.

The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A000201 as the parent: A000201, A001030, A001468, A001950, A003622, A003842, A003849, A004641, A005614, A014675, A022342, A088462, A096270, A114986, A124841. - N. J. A. Sloane, Mar 11 2021

[Floor((Sqrt(2)-1)*n+1/Sqrt(2)): n in [1..100]]; // Vincenzo Librandi, Jun 26 2017

Table[Floor[(Sqrt[2] - 1) n + 1 / Sqrt[2]], {n, 100}] (* Vincenzo Librandi, Jun 26 2017 *)

l=[0, 1, 1]
for n in range(3, 101): l.append(n - l[n - 1] - l[l[n - 2]])
print(l[1:]) # Indranil Ghosh, Jun 24 2017, after Altug Alkan

a(n) = floor((sqrt(2)-1)*n + 1/sqrt(2)).

a(1) = a(2) = 1; a(n) = n - a(n-1) - a(a(n-2)) for n > 2. - Altug Alkan, Jun 24 2017

A114986 Characteristic function of (A000201 prefixed with 0).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0

Offset: 0

N. J. A. Sloane, Feb 28 2006

Lord, E. A, Ranganathan, S. and Subramaniam, A., Stacking sequences and symmetry properties of trigonal vacancy-ordered phases, Phil. Mag. A 82 (2002) 255-268.
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)
Index entries for characteristic functions

Essentially the same as A005614. Cf. A096270, A189479.

A124842 Triangle, row sums = A005614, the rabbit sequence.

Original entry on oeis.org

1, 1, -1, 1, -2, 2, 1, -3, 6, -3, 1, -4, 12, -12, 3, 1, -5, 20, -30, 15, 0, 1, -6, 30, -60, 45, 0, -10, 1, -7, 42, -105, 105, 0, -70, 35, 1, -8, 56, -168, 210, 0, -280, 280, -90

Offset: 1

Gary W. Adamson, Nov 10 2006

			First few rows of the triangle are:
1;
1, -1;
1, -2, 2;
1, -3, 6, -3;
1, -4, 12, -12, 3;
1, -5, 20, -30, 15, 0;
1, -6, 30, -60, 45, 0, -10;
...
4th term of the rabbit sequence (1, 0, 1, 1, 0...) = 1 = sum of row 4 terms: (1, - 3 + 6 - 3).

Cf. A124841.

Binomial transform of the infinite matrix with the diagonalized sequence A124841.

cp's OEIS Frontend

A001030 Fixed under 1 -> 21, 2 -> 211.

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A004641 Fixed under 0 -> 10, 1 -> 100.

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A088462 a(1)=1, a(n) = ceiling((n - a(a(n-1)))/2).

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A114986 Characteristic function of (A000201 prefixed with 0).

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A124842 Triangle, row sums = A005614, the rabbit sequence.

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