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%I #39 Aug 17 2022 16:20:36
%S 1,2,4,5,7,9,10,12,13,15,17,18,20,22,23,25,26,28,30,31,33,34,36,38,39,
%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 78,80,81,83,85,86,88,89,91,93,94,96,98,99
%N a(n) = floor(n*phi) + 1, where phi = (1+sqrt(5))/2.
%C a(n)=least k such that s(k)=n, where s=A026350.
%C a(n)=position of n-th 1 in A096270.
%C From _Wolfdieter Lang_, Jun 27 2011: (Start)
%C a(n) = A(n)+1, with Wythoff sequence A(n)=A000201(n), n>=1, and A(0)=0.
%C a(n) = -floor(-n*phi). Recall that floor(-x) = -(floor(x)+1) if x is not integer and -floor(x) otherwise.
%C An exhaustive and disjoint decomposition of the integers is given by the following two Wythoff sequences A' and B: A'(0):=-1 (not 0), A'(-n):=-a(n)=-(A(n)+1), n>=1, A'(n) = A(n), n>=1, and B(-n):=-(B(n)+1)= -A026352(n), n>=1, with B(n)=A001950(n), n>=1, and B(0)=0.
%C (End)
%C Where odd terms in A060142 occur: A060142(a(n)) = A219608(n). - _Reinhard Zumkeller_, Nov 26 2012
%H Carmine Suriano, <a href="/A026351/b026351.txt">Table of n, a(n) for n = 0..10000</a>
%H B. Cloitre, N. J. A. Sloane and M. J. Vandermast, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.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.
%H J. H. Conway and N. J. A. Sloane, <a href="/A019586/a019586.pdf">Notes on the Para-Fibonacci and related sequences</a>
%H Eric Friedman, Scott M. Garrabrant, Ilona K. Phipps-Morgan, A. S. Landsberg and Urban Larsson, <a href="http://library.msri.org/books/Book70/files/1015.pdf">Geometric analysis of a generalized Wythoff game</a>, in Games of no Chance 5, MSRI publ. Cambridge University Press, 2019.
%H N. J. A. Sloane, <a href="/classic.html#WYTH">Classic Sequences</a>
%t Table[Floor[n*GoldenRatio] + 1, {n, 0, 100}] (* _T. D. Noe_, Apr 15 2011 *)
%o (Haskell)
%o import Data.List (findIndices)
%o a026351 n = a026351_list !! n
%o a026351_list = findIndices odd a060142_list
%o -- _Reinhard Zumkeller_, Nov 26 2012
%o (Python)
%o from math import isqrt
%o def A026351(n): return (n+isqrt(5*n**2)>>1)+1 # _Chai Wah Wu_, Aug 17 2022
%Y Essentially same as A004956. Cf. A000201.
%Y Complement of A026352.
%K nonn,easy,nice
%O 0,2
%A _N. J. A. Sloane_, _Clark Kimberling_
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