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A066096 a(n) = floor(n*phi), where phi = (1 + sqrt(5))/2.

%I A066096 #60 Aug 01 2025 17:05:25
%S A066096 0,1,3,4,6,8,9,11,12,14,16,17,19,21,22,24,25,27,29,30,32,33,35,37,38,
%T A066096 40,42,43,45,46,48,50,51,53,55,56,58,59,61,63,64,66,67,69,71,72,74,76,
%U A066096 77,79,80,82,84,85,87,88,90,92,93,95,97,98,100,101,103,105,106
%N A066096 a(n) = floor(n*phi), where phi = (1 + sqrt(5))/2.
%C A066096 a(n) is the smallest number different from a(i) and a(i)+i for i < n.
%C A066096 The losing positions in the game of Wythoff-Nim are precisely the pairs (a(n), a(n)+n).
%H A066096 G. C. Greubel, <a href="/A066096/b066096.txt">Table of n, a(n) for n = 0..10000</a>
%F A066096 For n >= 1, a(n) = A000201(n).
%F A066096 Duplicate values in A060143.
%F A066096 a(n) = 1 + A022342(n) = A000201(n).
%F A066096 a(n) = floor(n*phi), where phi = (1 + sqrt(5))/2. - _Peter Munn_, Jan 12 2018
%F A066096 a(n) = A026351(n) - 1. - _Philippe Deléham_, Jan 15 2023
%t A066096 Floor[GoldenRatio*Range[0, 80]] (* _G. C. Greubel_, Sep 12 2023 *)
%o A066096 (PARI) a(n) = (n+sqrtint(5*n^2))\2;
%o A066096 [a(n)|n<-[0..100]] \\ _Simon Strandgaard_, Jun 28 2022
%o A066096 (Magma) [Floor((1+Sqrt(5))*n/2): n in [0..80]]; // _G. C. Greubel_, Sep 12 2023
%o A066096 (SageMath) [floor(golden_ratio*n) for n in range(81)] # _G. C. Greubel_, Sep 12 2023
%Y A066096 Essentially the partial sums of A001468.
%Y A066096 Cf. A000201, A001622, A003622, A022342, A026351, A035336, A060143.
%Y A066096 Cf. A004919, A004920, A004921, A004922, A004923, A004924, A004925.
%Y A066096 Cf. A004926, A004927, A004928, A004929, A004930, A004931, A004932.
%Y A066096 Cf. A004933, A004934, A004935, A004976, A090909.
%K A066096 nonn,easy
%O A066096 0,3
%A A066096 Michele Dondi (bik.mido(AT)tiscalenet.it), Dec 30 2001
%E A066096 Name corrected by _Peter Munn_, Dec 06 2017
%E A066096 New name using a formula from _Peter Munn_ by _Peter Luschny_, Jan 18 2023