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%I #38 Aug 26 2022 06:25:53
%S 1,4,5,8,11,12,15,16,19,22,23,26,29,30,33,34,37,40,41,44,45,48,51,52,
%T 55,58,59,62,63,66,69,70,73,76,77,80,81,84,87,88,91,92,95,98,99,102,
%U 105,106,109,110,113,116,117,120,121,124,127,128,131,134,135,138,139
%N a(n) = 2*floor(n*phi)-n, where phi = (1+sqrt(5))/2.
%C Old name was a(n) = last number in repeating block in continued fraction for n*phi.
%D Alfred Brousseau, Fibonacci and Related Number Theoretic Tables, The Fibonacci Association, 1972, 101-103.
%H Clark Kimberling, <a href="/A050140/b050140.txt">Table of n, a(n) for n = 1..1000</a>
%H J.-P. Allouche and F. M. Dekking, <a href="https://arxiv.org/abs/1809.03424">Generalized Beatty sequences and complementary triples</a>, arXiv:1809.03424 [math.NT], 2018.
%F a(n) = -n + 2*floor(n*phi) = A283233(n)-n.
%F a(n) = floor(n*phi) + floor(n*sigma) where phi = (sqrt(5)+1)/2 and sigma = (sqrt(5)-1)/2.
%F a(n) = last number in repeating block in continued fraction for n*phi.
%t Table[-n+2Floor[n*GoldenRatio],{n,1,100}]
%o (PARI) for(n=1,50, print1(-n + 2*floor(n*(1+sqrt(5))/2), ", ")) \\ _G. C. Greubel_, Oct 15 2017
%o (Magma) [-n + 2*Floor(n*(1+Sqrt(5))/2): n in [1..50]]; // _G. C. Greubel_, Oct 15 2017
%o (Python)
%o def A050140(n): return (n+isqrt(5*n**2)&-2)-n # _Chai Wah Wu_, Aug 25 2022
%Y Cf. A000201, A001622, A005206, A050141, A005614, A001350.
%K nonn
%O 1,2
%A _Clark Kimberling_
%E Formula and more terms from _Vladeta Jovovic_, Nov 23 2001
%E Name changed by _Michel Dekking_, Dec 27 2017
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