

A080754


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


5



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


COMMENTS

Integer solutions >0 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 zeroone sequence A188037.  Clark Kimberling, Mar 19 2011


LINKS

G. C. Greubel, Table of n, a(n) for n = 1..5000
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, arXiv:math/0305308 [math.NT], 2003.


FORMULA

a(1) = 3; for n>1, a(n) = a(n1) + 3 if n is in sequence, a(n) = a(n1) + 2 if not.


MATHEMATICA

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


PROG

(PARI) for(n=1, 30, print1(ceil(n*(1+sqrt(2))), ", ")) \\ G. C. Greubel, Nov 28 2017
(MAGMA) [Ceiling(n*(1+Sqrt(2))): n in [1..30]]; // G. C. Greubel, Nov 28 2017


CROSSREFS

Equals A003151 + 1. This and its complement A080755 partition the integers >= 2.
Sequence in context: A033033 A211704 A275813 * A198083 A195168 A047218
Adjacent sequences: A080751 A080752 A080753 * A080755 A080756 A080757


KEYWORD

nonn


AUTHOR

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


EXTENSIONS

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


STATUS

approved



