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%I #10 Aug 20 2017 09:34:53
%S 1,3,6,7,11,12,13,18,19,20,21,27,28,29,30,31,38,39,40,41,42,43,51,52,
%T 53,54,55,56,57,66,67,68,69,70,71,72,73,83,84,85,86,87,88,89,90,91,
%U 102,103,104,105,106,107,108,109,110,111,123,124,125,126,127,128,129
%N Waterbird take-off sequence. Complement of A166021.
%C This kind of sequence can be generalized as follows:
%C Let F(t), G(t) be arithmetic functions: F(t) the right hand move, G(t) the number of erased positions.
%C Then starting from the position t=1 do procedure:
%C JUMP F(t) positions right hand
%C ERASE G(t) positions
%C SET t=t+1
%C repeat procedure from the last erased position.
%C This sequence has F(t)=t, G(t)=t.
%C We can use a set of functions F_i(t) and G_i(t) processed in parallel (a flock of birds taking off).
%H D. X. Charles, <a href="http://pages.cs.wisc.edu/~cdx/Sieve.pdf">Sieve Methods</a>, July 2000, University of Wisconsin.
%H R. Eismann, <a href="http://arxiv.org/abs/0711.0865">Decomposition of natural numbers into weight X level + jump and application to a new classification of prime numbers</a>, arXiv:0711.0865 [math.NT], 2007-2010.
%H M. C. Wunderlich, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa16/aa1614.pdf">A general class of sieve generated sequences</a>, Acta Arithmetica XVI,1969, pp. 41-56.
%F a(0)=1; let t=1. Start on position t. Jump t positions right hand. Erase t positions. (*P*) Set t=t+1. Start on the last erased position. Jump t positions right hand. Erase t positions. Repeat procedure (*P*).
%e 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,...
%e t=1; from the position 1 go 1 position to the right, erase 1 position:
%e 1..3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,...
%e t=2; from the last erased position go 2 positions to the right, erase 2 positions:
%e 1..3..,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,...
%e t=3; from the last erased position go 3 positions to the right, erase 3 positions:
%e 1..3..,6,7....11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,...
%e t=4; from the last erased position go 4 positions to the right, erase 4 positions:
%e 1..3..,6,7....11,12,13....,18,19,20,21,22,23,24,25,26,27,...
%e t=5; from the last erased position go 5 positions to the right, erase 5 positions:
%e 1..3..,6,7....11,12,13....,18,19,20,21......27,...
%e The erased positions form the complement of this sequence: A166021.
%Y Cf. A137292.
%K easy,nonn
%O 1,2
%A _Ctibor O. Zizka_, Mar 19 2008
%E Edited and corrected by _Antti Karttunen_, Oct 05 2009
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