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

%I #18 Sep 08 2022 08:45:55

%S 3,8,14,19,24,29,35,40,45,50,55,61,66,71,76,82,87,92,97,103,108,113,

%T 118,124,129,134,139,144,150,155,160,165,171,176,181,186,192,197,202,

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

%C See A184117 for the definition of lower and upper s-Wythoff sequences.

%H G. C. Greubel, <a href="/A184517/b184517.txt">Table of n, a(n) for n = 1..10000</a>

%H Clark Kimberling, <a href="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 a(n) = ceiling((2*n-1)*phi^2), where phi = A001622. - _Jon Maiga_, Nov 15 2018

%t k = 4; r = 2; d = Sqrt[4 + k^2];

%t a[n_] := Floor[(1/2) (d + 2 - k) (n + r/(d + 2))];

%t b[n_] := Floor[(1/2) (d + 2 + k) (n - r/(d + 2))];

%t Table[a[n], {n, 120}] (* A184516 *)

%t Table[b[n], {n, 120}] (* A184517 *)

%t (* alternate program *)

%t Table[Ceiling[(2 n - 1) GoldenRatio^2], {n, 1, 120}] (* _Jon Maiga_, Nov 15 2018 *)

%o (PARI) vector(100, n, floor((3+sqrt(5))*(n - 1/(1+sqrt(5))))) \\ _G. C. Greubel_, Nov 16 2018

%o (Magma) [Floor((3+Sqrt(5))*(n - 1/(1+Sqrt(5)))): n in [1..100]]; // _G. C. Greubel_, Nov 16 2018

%o (Sage) [floor((3+sqrt(5))*(n - 1/(1+sqrt(5)))) for n in (1..100)] # _G. C. Greubel_, Nov 16 2018

%Y Cf. A184117, A184516.

%K nonn

%O 1,1

%A _Clark Kimberling_, Jan 16 2011

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Last modified March 28 20:05 EDT 2024. Contains 371254 sequences. (Running on oeis4.)