%I #28 Sep 08 2022 08:46:12
%S 1,2,4,5,7,8,9,11,12,14,15,16,18,19,21,22,23,25,26,28,29,31,32,33,35,
%T 36,38,39,40,42,43,45,46,48,49,50,52,53,55,56,57,59,60,62,63,64,66,67,
%U 69,70,72,73,74,76,77,79,80,81,83,84,86,87,89,90,91,93
%N Nonhomogeneous Beatty sequence: ceiling((n + 1/4)*sqrt(2)).
%C Complement of A258834.
%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="/A258833/b258833.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)*sqrt(2)) = floor((n + 1/4)*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)*Sqrt(2)): n in [0..80]]; // _Vincenzo Librandi_, Jun 13 2015
%o (PARI) for(n=0,50, print1(ceil((n + 1/4)*sqrt(2)), ", ")) \\ _G. C. Greubel_, Feb 08 2018
%Y Cf. A258834 (complement), A184580, A184581.
%K nonn,easy
%O 0,2
%A _Clark Kimberling_, Jun 12 2015