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%I #34 Feb 14 2024 17:27:25
%S 3,5,8,10,13,15,17,20,22,25,27,29,32,34,37,39,42,44,46,49,51,54,56,58,
%T 61,63,66,68,71,73,75,78,80,83,85,87,90,92,95,97,99,102,104,107,109,
%U 112,114,116,119,121,124,126,128,131,133,136,138,141,143,145
%N a(n) = ceiling(n*(1+sqrt(2))).
%C Positive integer solutions to the equation x = ceiling(r*floor(x/r)), where r = 1+sqrt(2). - _Benoit Cloitre_, Feb 14 2004
%C Equivalently, numbers m such that {rm} <= {r}, where r=2^(1/2) and { } denotes fractional part.
%C _Andrew Plewe_, May 18 2007, observed that the sequence defined by a(n) = ceiling(n*(1+sqrt(2))) appeared to give the same numbers as the sequence, originally due to _Clark Kimberling_, Jul 01 2006, defined by: numbers m such that {rm} <= {r}, where r=2^(1/2). That these sequences are indeed the same was shown by _David Applegate_. This follows since the complements of the two sequences are the same, which is shown in the comments on A080755.
%C It appears that A080754 gives the positions of 1 in the zero-one sequence A188037. - _Clark Kimberling_, Mar 19 2011
%H G. C. Greubel, <a href="/A080754/b080754.txt">Table of n, a(n) for n = 1..5000</a>
%H Benoit Cloitre, N. J. A. Sloane, and Matthew J. Vandermast, <a href="http://www.emis.de/journals/JIS/VOL6/Cloitre/cloitre2.html">Numerical analogues of Aronson's sequence</a>, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
%H Benoit Cloitre, N. J. A. Sloane, and Matthew J. Vandermast, <a href="https://arxiv.org/abs/math/0305308">Numerical analogues of Aronson's sequence</a>, arXiv:math/0305308 [math.NT], 2003.
%H Luke Schaeffer, Jeffrey Shallit, and Stefan Zorcic, <a href="https://arxiv.org/abs/2402.08331">Beatty Sequences for a Quadratic Irrational: Decidability and Applications</a>, arXiv:2402.08331 [math.NT], 2024. See pp. 17-19.
%F a(1) = 3; for n>1, a(n) = a(n-1) + 3 if n is in sequence, a(n) = a(n-1) + 2 if not.
%t Table[Ceiling[n*(1 + Sqrt[2])], {n, 1, 50}] (* _G. C. Greubel_, Nov 28 2017 *)
%o (PARI) for(n=1,30, print1(ceil(n*(1+sqrt(2))), ", ")) \\ _G. C. Greubel_, Nov 28 2017
%o (Magma) [Ceiling(n*(1+Sqrt(2))): n in [1..30]]; // _G. C. Greubel_, Nov 28 2017
%Y Equals A003151 + 1. This and its complement A080755 partition the integers >= 2.
%K nonn
%O 1,1
%A _Benoit Cloitre_ and _N. J. A. Sloane_, Mar 09 2003
%E Edited by _N. J. A. Sloane_ at the suggestion of _Andrew S. Plewe_, Jun 08 2007
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