%I #53 Aug 01 2024 23:15:55
%S 0,2,4,5,7,9,10,12,13,15,17,18,20,22,23,25,26,28,30,31,33,34,36,38,39,
%T 41,43,44,46,47,49,51,52,54,56,57,59,60,62,64,65,67,68,70,72,73,75,77,
%U 78,80,81,83,85,86,88,89
%N a(n) = ceiling(n*phi), where phi is the golden ratio, A001622.
%C a(0)=0, a(1)=2; for n > 1, a(n) = a(n-1) + 2 if n is already in the sequence, a(n) = a(n-1) + 1 otherwise.
%C Integer solutions to the equation x = ceiling(phi*floor(x/phi)). - _Benoit Cloitre_, Feb 14 2004
%C From _Benoit Cloitre_, Mar 05 2007: (Start)
%C The following is an alternative way to obtain this sequence. NP means "term not in parentheses". Write down the natural numbers and mark the least NP, which is 1:
%C 1* 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 ...
%C Take the first NP (which is 1) and parenthesize it; mark the least NP (which is 2):
%C (1*) 2* 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 ...
%C Take the 2nd NP (which is 3) and parenthesize it; mark the next NP (which is 4):
%C (1*) 2* (3) 4* 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 ...
%C Take the 4th NP (which is 6) and parenthesize it; mark the next NP (which is 5):
%C (1*) 2* (3) 4* 5* (6) 7 8 9 10 11 12 13 14 15 16 17 18 19 ...
%C Continuing in this way we obtain
%C (1*) 2* (3) 4* 5* (6) 7* (8) 9* 10* (11) 12* 13* (14) 15* (16) 17* (18) 19* ...
%C The starred entries (after the first) give the sequence. (End)
%C From _Rick L. Shepherd_, Dec 05 2009: (Start)
%C An equivalent statement of the sieving process described by _Benoit Cloitre_ on Mar 05 2007:
%C Begin with the natural numbers N. Repeatedly perform these two steps:
%C i) Let k = N's least remaining term not yet used in Step ii).
%C ii) Remove the k-th remaining term from N.
%C The remaining terms of N are the (positive) terms shared by this sequence and A026351.
%C The terms removed from N (the complement) are A026352's terms (see also A004957).
%C The PARI program performs this sieving process and prints the positive terms of this sequence. (End)
%H Christian Krause, <a href="/A004956/b004956.txt">Table of n, a(n) for n = 0..10000</a>
%H B. Cloitre, N. J. A. Sloane and M. J. Vandermast, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL6/Cloitre/cloitre2.html">Numerical analogues of Aronson's sequence</a>, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
%H B. Cloitre, N. J. A. Sloane and M. J. Vandermast, <a href="https://arxiv.org/abs/math/0305308">Numerical analogues of Aronson's sequence</a>, arXiv:math/0305308 [math.NT], 2003.
%t Ceiling[Range[0,100]GoldenRatio] (* _Paolo Xausa_, Oct 28 2023 *)
%o (PARI) {/* paws = Print Absolute values of all elements in vector v With same Sign as sn */
%o paws(v,sn) = for(m=1,matsize(v)[2], if(sign(v[m])==sign(sn),\
%o print1(abs(v[m]),",")))}
%o {/* Sieve through lim numbers; make values negative to signify "removed" */
%o lim=100; v=vector(lim,i,i); i=0; j=0; c=1;
%o while(i<lim, i++; if(v[i]>0, k=v[i]; c=c--;
%o while(c<k && j<lim, j++; if(v[j]>0, c++)); v[j]=-v[j])); paws(v,1)\
%o /* Changing "1" to "-1" in paws() above prints out the terms of A026352 instead */} \\ _Rick L. Shepherd_, Dec 05 2009
%o (PARI) a(n) = ceil(n*(1 + sqrt(5))/2); \\ _Michel Marcus_, Apr 13 2021
%Y Cf. A001622, A004957, A026352.
%Y Essentially same as A026351.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_