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%I #53 Sep 08 2022 08:45:26
%S 0,1,1,3,4,4,6,6,8,9,9,11,12,12,14,14,16,17,17,19,19,21,22,22,24,25,
%T 25,27,27,29,30,30,32,33,33,35,35,37,38,38,40,40,42,43,43,45,46,46,48,
%U 48,50,51,51,53,53,55,56,56,58,59,59,61,61,63,64,64,66,67,67,69,69,71,72
%N a(n) = floor(phi*floor(n/phi)) where phi=(1+sqrt(5))/2.
%H N. J. A. Sloane, <a href="/A120613/b120613.txt">Table of n, a(n) for n = 1..10000</a> (corrected by Michel Dekking, uploaded again by _Georg Fischer_, Jan 31 2019)
%H F. Michel Dekking, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Dekking/dekk4.html">Morphisms, Symbolic Sequences, and Their Standard Forms</a>, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.
%H Martin Griffiths, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.118.06.497">The Golden String, Zeckendorf Representations, and the Sum of a Series</a>, Amer. Math. Monthly, 118 (2011), 497-507. See p. 502.
%F a(n) = n - A003842(n-2) for n >= 2. [Corrected by _Georg Fischer_, Jan 31 2019]
%F In particular, a(n) = n-1 or a(n) = n-2. - _Charles R Greathouse IV_, Aug 26 2022
%t Table[Floor[GoldenRatio*Floor[n/GoldenRatio]], {n,1,100}] (* _G. C. Greubel_, Oct 23 2018 *)
%o (PARI) f=(1+sqrt(5))/2;a(n)=floor(f*floor(n/f))
%o (Magma) [Floor((1+Sqrt(5))*Floor(2*n/(1+Sqrt(5)))/2): n in [1..100]]; // _G. C. Greubel_, Oct 23 2018
%o (Python)
%o from math import isqrt
%o def A120613(n): return (m:=(n+isqrt(5*n**2)>>1)-n)+isqrt(5*m**2)>>1 # _Chai Wah Wu_, Aug 26 2022
%Y Cf. A001622, A120614 (first differences), A120615 (partial sums), A003842.
%K nonn
%O 1,4
%A _Benoit Cloitre_, Jun 17 2006
%E Offset changed by _Michel Dekking_, Oct 23 2018
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