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

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

1,1

Comments

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

Links

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

Formula

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)

Maple

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

Mathematica

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 *)

Prog

(Magma) [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
(Python)
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

Crossrefs

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

Sequence in context: A189298 A288375 A121559 * A266441 A266672 A266070

Adjacent sequences: A004638 A004639 A004640 * A004642 A004643 A004644

Keyword

nonn,nice,easy

Author

N. J. A. Sloane

Status

approved