# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/

Search: id:a004641
Showing 1-1 of 1

%I A004641 #74 Aug 26 2022 10:26:26
%S A004641 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,
%T A004641 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,
%U A004641 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
%N A004641 Fixed under 0 -> 10, 1 -> 100.
%C A004641 Partial sums: A088462. - _Reinhard Zumkeller_, Dec 05 2009
%C A004641 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
%H A004641 T. D. Noe, <a href="/load/view.php?a=aHR0cDovL29laXMub3JnL0EwMDQ2NDEvYjAwNDY0MS50eHQ">Table of n, a(n) for n = 1..8119</a>
%H A004641 Wieb Bosma, Michel Dekking, and Wolfgang Steiner, <a href="/load/view.php?a=aHR0cHM6Ly9hcnhpdi5vcmcvYWJzLzE3MTAuMDE0OTg">A remarkable sequence related to Pi and sqrt(2)</a>, arXiv:1710.01498 [math.NT], 2017.
%H A004641 Wieb Bosma, Michel Dekking, and Wolfgang Steiner, <a href="/load/view.php?a=aHR0cDovL21hdGguY29sZ2F0ZS5lZHUvfmludGVnZXJzL3NqczQvc2pzNC5BYnN0cmFjdC5odG1s">A remarkable sequence related to Pi and sqrt(2)</a>, Integers, Electronic Journal of Combinatorial Number Theory 18A (2018), #A4.
%H A004641 N. G. de Bruijn, <a href="/load/view.php?a=aHR0cDovL2R4LmRvaS5vcmcvMTAuMTAxNi8xMzg1LTcyNTgoODEpOTAwMTUtOQ">Sequences of zeros and ones generated by special production rules</a>, Nederl. Akad. Wetensch. Indag. Math. 43 (1981), no. 1, 27-37. Reprinted in Physics of Quasicrystals, ed. P. J. Steinhardt et al., p. 664.
%H A004641 C. J. Glasby, S. P. Glasby, and F. Pleijel, <a href="/load/view.php?a=aHR0cDovL2R4LmRvaS5vcmcvMTAuMTA5OC9yc3BiLjIwMDguMDQxOA">Worms by number</a>, Proc. Roy. Soc. B, Proc. Biol. Sci. 275 (1647) (2008) 2071-2076.
%H A004641 N. J. A. Sloane, <a href="/load/view.php?a=aHR0cDovL29laXMub3JnL0ExMTUwMDQvYTExNTAwNC50eHQ">Families of Essentially Identical Sequences</a>, Mar 24 2021 (Includes this sequence).
%H A004641 <a href="/load/view.php?a=aHR0cDovL29laXMub3JnL2luZGV4L0NoI2NoYXJfZm5z">Index entries for characteristic functions</a>
%F A004641 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
%F A004641 From _Jianing Song_, Jan 02 2019: (Start)
%F A004641 a(n) = A001030(n) - 1.
%F A004641 a(n) = A006337(n-9) - 1 = A159684(n-10) for n >= 10. (End)
%p A004641 P(0):= (1,0): P(1):= (1,0,0):
%p A004641 ((P~)@@6)([1]);
%p A004641 # in Maple 12 or earlier, comment the above line and uncomment the following:
%p A004641 # (curry(map,P)@@6)([1]); # _Robert Israel_, Mar 26 2015
%t A004641 Nest[ Flatten[# /. {0 -> {1, 0}, 1 -> {1, 0, 0}}] &, {1}, 5] (* _Robert G. Wilson v_, May 25 2011 *)
%t A004641 SubstitutionSystem[{0->{1,0},1->{1,0,0}},{1},5]//Flatten (* _Harvey P. Dale_, Nov 20 2021 *)
%o A004641 (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
%o A004641 (Python)
%o A004641 from math import isqrt
%o A004641 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
%Y A004641 Equals A001030 - 1. Essentially the same as A006337 - 1 and A159684.
%Y A004641 Characteristic function of A086377.
%Y A004641 Cf. A081477.
%Y A004641 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
%K A004641 nonn,nice,easy
%O A004641 1,1
%A A004641 _N. J. A. Sloane_

# Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE