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A136272
Waterbird take-off sequence. Complement of A166021.
10
1, 3, 6, 7, 11, 12, 13, 18, 19, 20, 21, 27, 28, 29, 30, 31, 38, 39, 40, 41, 42, 43, 51, 52, 53, 54, 55, 56, 57, 66, 67, 68, 69, 70, 71, 72, 73, 83, 84, 85, 86, 87, 88, 89, 90, 91, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 123, 124, 125, 126, 127, 128, 129
OFFSET
1,2
COMMENTS
This kind of sequence can be generalized as follows:
Let F(t), G(t) be arithmetic functions: F(t) the right hand move, G(t) the number of erased positions.
Then starting from the position t=1 do procedure:
JUMP F(t) positions right hand
ERASE G(t) positions
SET t=t+1
repeat procedure from the last erased position.
This sequence has F(t)=t, G(t)=t.
We can use a set of functions F_i(t) and G_i(t) processed in parallel (a flock of birds taking off).
LINKS
D. X. Charles, Sieve Methods, July 2000, University of Wisconsin.
M. C. Wunderlich, A general class of sieve generated sequences, Acta Arithmetica XVI,1969, pp. 41-56.
FORMULA
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*).
EXAMPLE
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,...
t=1; from the position 1 go 1 position to the right, erase 1 position:
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,...
t=2; from the last erased position go 2 positions to the right, erase 2 positions:
1..3..,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,...
t=3; from the last erased position go 3 positions to the right, erase 3 positions:
1..3..,6,7....11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,...
t=4; from the last erased position go 4 positions to the right, erase 4 positions:
1..3..,6,7....11,12,13....,18,19,20,21,22,23,24,25,26,27,...
t=5; from the last erased position go 5 positions to the right, erase 5 positions:
1..3..,6,7....11,12,13....,18,19,20,21......27,...
The erased positions form the complement of this sequence: A166021.
PROG
(PARI) is_A136272(n) = if(n<3, n==1, my(s=sqrtint(n-2)); n-2 < s*(s+1)) \\ Hugo Pfoertner, Jul 19 2024
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Ctibor O. Zizka, Mar 19 2008
EXTENSIONS
Edited and corrected by Antti Karttunen, Oct 05 2009
STATUS
approved