A080754 a(n) = ceiling(n*(1+sqrt(2))).

3, 5, 8, 10, 13, 15, 17, 20, 22, 25, 27, 29, 32, 34, 37, 39, 42, 44, 46, 49, 51, 54, 56, 58, 61, 63, 66, 68, 71, 73, 75, 78, 80, 83, 85, 87, 90, 92, 95, 97, 99, 102, 104, 107, 109, 112, 114, 116, 119, 121, 124, 126, 128, 131, 133, 136, 138, 141, 143, 145

Offset: 1

Author

Benoit Cloitre and N. J. A. Sloane, Mar 09 2003

Keywords

• nonn

Comments

Positive integer solutions to the equation x = ceiling(r*floor(x/r)), where r = 1+sqrt(2). - Benoit Cloitre, Feb 14 2004

Equivalently, numbers m such that {rm} <= {r}, where r=2^(1/2) and { } denotes fractional part.

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.

It appears that A080754 gives the positions of 1 in the zero-one sequence A188037. - Clark Kimberling, Mar 19 2011

Links

• G. C. Greubel, Table of n, a(n) for n = 1..5000
• Benoit Cloitre, A study of a family of self-referential sequences, arXiv:2506.18103 [math.GM], 2025. See p. 7.
• Benoit Cloitre, N. J. A. Sloane, and Matthew J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
• Benoit Cloitre, N. J. A. Sloane, and Matthew J. Vandermast, Numerical analogues of Aronson's sequence, arXiv:math/0305308 [math.NT], 2003.
• Luke Schaeffer, Jeffrey Shallit, and Stefan Zorcic, Beatty Sequences for a Quadratic Irrational: Decidability and Applications, arXiv:2402.08331 [math.NT], 2024. See pp. 17-19.

Crossrefs

Equals A003151 + 1. This and its complement A080755 partition the integers >= 2.

Programs

Magma

[Ceiling(n*(1+Sqrt(2))): n in [1..30]]; // G. C. Greubel, Nov 28 2017

Mathematica

Table[Ceiling[n*(1 + Sqrt[2])], {n, 1, 50}] (* G. C. Greubel, Nov 28 2017 *)

PARI

for(n=1,30, print1(ceil(n*(1+sqrt(2))), ", ")) \\ G. C. Greubel, Nov 28 2017

Formula

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.

Extensions

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 08 2007