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A003151 Beatty sequence for 1+sqrt(2); a(n) = floor(n*(1+sqrt(2))).
(Formerly M1033)
39
2, 4, 7, 9, 12, 14, 16, 19, 21, 24, 26, 28, 31, 33, 36, 38, 41, 43, 45, 48, 50, 53, 55, 57, 60, 62, 65, 67, 70, 72, 74, 77, 79, 82, 84, 86, 89, 91, 94, 96, 98, 101, 103, 106, 108, 111, 113, 115, 118, 120, 123, 125, 127, 130, 132, 135, 137, 140, 142, 144 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
a(1)=2; for n>1, a(n+1)=a(n)+3 if n is already in the sequence, a(n+1)=a(n)+2 otherwise.
Numbers with an odd number of trailing 0's in their minimal representation in terms of the positive Pell numbers (A317204). - Amiram Eldar, Mar 16 2022
From Clark Kimberling, Dec 24 2022: (Start)
This is the first of four sequences that partition the positive integers. Starting with a general overview, suppose that u = (u(n)) and v = (v(n)) are increasing sequences of positive integers. Let u' and v' be their complements, and assume that the following four sequences are infinite:
(1) u ^ v = intersection of u and v (in increasing order);
(2) u ^ v';
(3) u' ^ v;
(4) u' ^ v'.
Every positive integer is in exactly one of the four sequences.
For A003151, u, v, u', v', are the Beatty sequences given by u(n) = floor(n*sqrt(2)) and v(n) = floor((1+sqrt(2))/2)*n), so that r = sqrt(2), s = (1+sqrt(2))/2, r' = (2+sqrt(2))/2, s' = 1 + 1/sqrt(2).
Assume that if w is any of the sequences u, v, u', v', then lim_{n->oo) w(n)/n exists and defines the (limiting) density of w. For w = u,v,u',v', denote the densities by r,s,r',s'. Then the densities of sequences (1)-(4) exist, and
1/(r*r') + 1/(r*s') + 1/(s*s') + 1/(s*r') = 1.
(1) u ^ v = (2, 4, 7, 9, 12, 14, 16, 19, 21, 24, 26, 28, 31, 33, ...) = A003151
(2) u ^ v' = (1, 5, 8, 11, 15, 18, 22, 25, 29, 32, 35, 39, 42, ...) = A001954
(3) u' ^ v = (284, 287, 289, 292, 294, 296, 299, 301, 304, 306, ...) = A356135
(4) u' ^ v' = (3, 6, 10, 13, 17, 20, 23, 27, 30, 34, 37, 40, 44, ...) = A003152
For results of compositions instead of intersections, see A184922. (End)
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Shiri Artstein-Avidan, Aviezri S. Fraenkel and Vera T. Sós, A two-parameter family of an extension of Beatty sequences, Discrete Math., Vol. 308, No. 20 (2008), pp. 4578-4588; preprint.
L. Carlitz, R. Scoville and V. E. Hoggatt, Jr., Pellian representatives, Fib. Quart., Vol. 10, No. 5 (1972), pp. 449-488.
Benoit Cloitre, N. J. A. Sloane, and Matthew J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seq., Vol. 6 (2003), Article 03.2.2; arXiv preprint, arXiv:math/0305308 [math.NT], 2003.
Joshua N. Cooper and Alexander W. N. Riasanovsky, On the Reciprocal of the Binary Generating Function for the Sum of Divisors, J. Int. Seq., Vol. 16 (2013), Article 13.1.8; preprint, 2012.
R. J. Mathar, Graphical representation among sequences closely related to this one (cf. N. J. A. Sloane, "Families of Essentially Identical Sequences").
Luke Schaeffer, Jeffrey Shallit, and Stefan Zorcic, Beatty Sequences for a Quadratic Irrational: Decidability and Applications, arXiv:2402.08331 [math.NT], 2024. See pp. 17-18.
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence).
MATHEMATICA
Table[Floor[n*(1 + Sqrt[2])], {n, 1, 50}] (* G. C. Greubel, Jul 02 2017 *)
PROG
(PARI) for(n=1, 50, print1(floor(n*(1 + sqrt(2))), ", ")) \\ G. C. Greubel, Jul 02 2017
(Python)
from math import isqrt
def A003151(n): return n+isqrt(n*n<<1) # Chai Wah Wu, Aug 03 2022
CROSSREFS
Complement of A003152.
Equals A001951(n) + n.
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
Bisections: A197878, A215247.
Sequence in context: A283964 A330085 A175884 * A189939 A219637 A189470
KEYWORD
nonn,easy
AUTHOR
STATUS
approved

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Last modified July 18 13:04 EDT 2024. Contains 374378 sequences. (Running on oeis4.)