%I M2392 #59 Feb 08 2024 16:12:19

%S 1,3,5,6,8,10,11,13,15,17,18,20,22,23,25,27,29,30,32,34,35,37,39,40,

%T 42,44,46,47,49,51,52,54,56,58,59,61,63,64,66,68,69,71,73,75,76,78,80,

%U 81,83,85,87,88,90,92,93,95,97,99,100,102,104,105,107,109,110,112,114,116

%N A Beatty sequence: a(n) = floor(n*(1+1/sqrt(2))).

%C Numbers with an even number of trailing 0's in their minimal representation in terms of the positive Pell numbers (A317204). - _Amiram Eldar_, Mar 16 2022

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H G. C. Greubel, <a href="/A003152/b003152.txt">Table of n, a(n) for n = 1..10000</a>

%H L. Carlitz, R. Scoville, and V. E. Hoggatt, Jr. <a href="http://www.fq.math.ca/Scanned/10-5/carlitz1.pdf">Pellian representations</a>, Fibonacci Quarterly, Vol. 10, No. 5 (1972), pp. 449-488.

%H Joshua N. Cooper and Alexander W. N. Riasanovsky, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Cooper/cooper3.html">On the Reciprocal of the Binary Generating Function for the Sum of Divisors</a>, J. Int. Seq., Vol. 16 (2013), Article 13.1.8; <a href="http://www.math.sc.edu/~cooper/Sigma.pdf">preprint</a>, 2012.

%H N. J. A. Sloane, <a href="/A115004/a115004.txt">Families of Essentially Identical Sequences</a>, Mar 24 2021 (Includes this sequence).

%H <a href="/index/Be#Beatty">Index entries for sequences related to Beatty sequences</a>.

%p Digits := 100: t := evalf(1+sin(Pi/4)): A:= n->floor(t*n): seq(floor((t*n)),n=1..68); # _Zerinvary Lajos_, Mar 27 2009

%t Table[Floor[n (1 + 1/Sqrt[2])], {n, 70}] (* _Vincenzo Librandi_, Dec 26 2015 *)

%o (Magma) [Floor(n*(1+1/Sqrt(2))): n in [1..70]]; // _Vincenzo Librandi_, Dec 26 2015

%o (PARI) a(n)=n+sqrtint(2*n^2)\2 \\ _Charles R Greathouse IV_, Jan 25 2022

%Y Complement of A003151.

%Y Cf. A109250, A317204.

%Y The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A003151 as the parent: A003151, A001951, A001952, A003152, A006337, A080763, A082844 (conjectured), A097509, A159684, A188037, A245219 (conjectured), A276862. - _N. J. A. Sloane_, Mar 09 2021

%Y Bisections: A001952, A001954.

%K nonn,easy

%O 1,2

%A _N. J. A. Sloane_