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A184517 Upper s-Wythoff sequence, where s=4n-2. Complement of A184516.

%I A184517 #19 Jan 05 2025 19:51:39
%S A184517 3,8,14,19,24,29,35,40,45,50,55,61,66,71,76,82,87,92,97,103,108,113,
%T A184517 118,124,129,134,139,144,150,155,160,165,171,176,181,186,192,197,202,
%U A184517 207,213,218,223,228,234,239,244,249,254,260,265,270,275,281,286,291,296,302,307,312,317,323,328,333,338,343,349,354,359,364,370,375,380,385,391,396,401,406,412,417,422,427,432,438,443,448,453,459,464,469,474,480,485,490,495,501,506
%N A184517 Upper s-Wythoff sequence, where s=4n-2.  Complement of A184516.
%C A184517 See A184117 for the definition of lower and upper s-Wythoff sequences.
%H A184517 G. C. Greubel, <a href="/A184517/b184517.txt">Table of n, a(n) for n = 1..10000</a>
%H A184517 Clark Kimberling, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/49-3/KimberlingGeneralized.pdf">Beatty Sequences and Wythoff Sequences, Generalized</a>, Fibonacci Quart. 49 (2011), no. 3, 195-200.
%F A184517 a(n) = ceiling((2*n-1)*phi^2), where phi = A001622. - _Jon Maiga_, Nov 15 2018
%t A184517 k = 4; r = 2; d = Sqrt[4 + k^2];
%t A184517 a[n_] := Floor[(1/2) (d + 2 - k) (n + r/(d + 2))];
%t A184517 b[n_] := Floor[(1/2) (d + 2 + k) (n - r/(d + 2))];
%t A184517 Table[a[n], {n, 120}] (* A184516 *)
%t A184517 Table[b[n], {n, 120}] (* A184517 *)
%t A184517 (* alternate program *)
%t A184517 Table[Ceiling[(2 n - 1) GoldenRatio^2], {n, 1, 120}] (* _Jon Maiga_, Nov 15 2018 *)
%o A184517 (PARI) vector(100, n, floor((3+sqrt(5))*(n - 1/(1+sqrt(5))))) \\ _G. C. Greubel_, Nov 16 2018
%o A184517 (Magma) [Floor((3+Sqrt(5))*(n - 1/(1+Sqrt(5)))): n in [1..100]]; // _G. C. Greubel_, Nov 16 2018
%o A184517 (Sage) [floor((3+sqrt(5))*(n - 1/(1+sqrt(5)))) for n in (1..100)] # _G. C. Greubel_, Nov 16 2018
%Y A184517 Cf. A184117, A184516.
%K A184517 nonn
%O A184517 1,1
%A A184517 _Clark Kimberling_, Jan 16 2011