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Nonhomogeneous Beatty sequence: ceiling((n - 1/4)*(2 + sqrt(2))).
2

%I #27 Dec 13 2023 08:47:32

%S 0,3,6,10,13,17,20,24,27,30,34,37,41,44,47,51,54,58,61,65,68,71,75,78,

%T 82,85,88,92,95,99,102,105,109,112,116,119,123,126,129,133,136,140,

%U 143,146,150,153,157,160,164,167,170,174,177,181,184,187,191,194

%N Nonhomogeneous Beatty sequence: ceiling((n - 1/4)*(2 + sqrt(2))).

%C Complement of A258833.

%C Let r = sqrt(2) and s = r/(r-1) = 2 + sqrt(2). Let R be the ordered set {floor[(n + 1/4)*r] : n is an integer} and let S be the ordered set {floor[(n - 1/4)*s : n is an integer}; thus,

%C R = (..., -8, -7, -5, -4, -2, -1, 1, 2, 3, 5, 6, ...)

%C S = (..., -13, -10, -6, -3, 0, 4, 7, 11, 14, ...).

%C By Fraenkel's theorem (Theorem XI in the cited paper); R and S partition the integers.

%C A184580 = (1,2,3,5,6,...), positive terms of R;

%C A184581 = (4,7,11,14,...), positive terms of S;

%C A258833 = (1,2,4,5,6,...), - (negative terms of R);

%C A258834 = (0,3,6,10,...), - (nonpositive terms of S).

%C A184580 and A184581 partition the positive integers, and A258833 and A248834 partition the nonnegative integers.

%H Clark Kimberling, <a href="/A258834/b258834.txt">Table of n, a(n) for n = 0..10000</a>

%H A. S. Fraenkel, <a href="http://dx.doi.org/10.4153/CJM-1969-002-7">The bracket function and complementary sets of integers</a>, Canadian J. of Math. 21 (1969) 6-27.

%H Clark Kimberling, <a href="https://www.emis.de/journals/INTEGERS/papers/q15/q15.Abstract.html">Beatty sequences and trigonometric functions</a>, Integers 16 (2016), #A15.

%F a(n) = ceiling((n - 1/4)*(2 + sqrt(2))) = floor((n - 1/4)*(2 + sqrt(2)) + 1).

%t r = Sqrt[2]; s = r/(r - 1);

%t Table[Ceiling[(n + 1/4) r], {n, 0, 100}] (* A258833 *)

%t Table[Ceiling[(n - 1/4) s], {n, 0, 100}] (* A258834 *)

%o (Magma) [Ceiling((n-1/4)*(2+Sqrt(2))): n in [0..80]]; // _Vincenzo Librandi_, Jun 13 2015

%o (PARI) vector(60, n, ceil((n-1/4)*(2+sqrt(2)))) \\ _G. C. Greubel_, Aug 19 2018

%Y Cf. A258833 (complement), A184580, A184581.

%K nonn,easy

%O 0,2

%A _Clark Kimberling_, Jun 12 2015

%E Corrected by _Michel Dekking_, Sep 19 2019.