A080755 -id:A080755 - cp's OEIS frontend

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

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

nonn

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

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.

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

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

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

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

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.

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

A083088 First column of table A083087.

Original entry on oeis.org

1, 2, 4, 6, 7, 9, 11, 12, 14, 16, 18, 19, 21, 23, 24, 26, 28, 30, 31, 33, 35, 36, 38, 40, 41, 43, 45, 47, 48, 50, 52, 53, 55, 57, 59, 60, 62, 64, 65, 67, 69, 70, 72, 74, 76, 77, 79, 81, 82, 84, 86, 88, 89, 91, 93, 94, 96, 98, 100, 101, 103, 105, 106, 108, 110, 111, 113, 115

Offset: 0

Paul D. Hanna, Apr 21 2003

It appears that A188937 gives the positions of 0 in the zero-one sequence A188037; complement of A080754. - Clark Kimberling, Mar 19 2011

Is this (apart from the prefixed a(0)) the same as A080755? - R. J. Mathar, Jul 31 2025

Cf. A083087, A083044, A083047, A083050.

Cf. A080755.

z:=70; x:=1+Sqrt(2); [ Floor(n*x/(x-1))+1: n in [0..z] ]; // Klaus Brockhaus, Jan 04 2011

f[n_] := Floor[n/Sqrt@2 + n + 1]; Array[f, 68, 0]

a(n) = floor(n*x/(x-1)) + 1, n>=0, where x=1+sqrt(2).

a(n) = floor(n/sqrt(2)) + n + 1 = 1+n+A049472(n).

This entry formerly contained an erroneous comment, which was deleted by N. J. A. Sloane, Jan 30 2008

cp's OEIS Frontend

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

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