A087057 Smallest number whose square is larger than 2*n^2.

2, 3, 5, 6, 8, 9, 10, 12, 13, 15, 16, 17, 19, 20, 22, 23, 25, 26, 27, 29, 30, 32, 33, 34, 36, 37, 39, 40, 42, 43, 44, 46, 47, 49, 50, 51, 53, 54, 56, 57, 58, 60, 61, 63, 64, 66, 67, 68, 70, 71, 73, 74, 75, 77, 78, 80, 81, 83, 84, 85, 87, 88, 90, 91, 92, 94, 95, 97, 98, 99, 101

Offset

1,1

Comments

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

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Formula

a(n) = 1 + A001951(n) = 1 + floor(n*sqrt(2)) = sqrt(A087058(n)).

a(n) = ceiling(n*sqrt(2)). - Vincenzo Librandi, Oct 22 2011

Example

a(10) = 15 because the 15^2 = 225 is the smallest square number greater than 2*10^2 = 200.
Can be built by recursive removals:
start with 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, ...
get a(1) := 2 and remove the 2nd term (= 4):
[2] _ 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, ...
get a(2) := 3 and remove the 3rd term (= 7):
[2, 3] _ 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, ...
get a(3) := 5 and remove the 5th term (= 11):
[2, 3, 5] _ 6, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, ...
get a(4) := 6 and remove the 6th term (= 14):
[2, 3, 5, 6] _ 8, 9, 10, 12, 13, 15, 16, 17, 18, 19, 20, 21, 22, 23, ...
get a(5) := 8 and remove the 8th term (= 18):
[2, 3, 5, 6, 8] _ 9, 10, 12, 13, 15, 16, 17, 19, 20, 21, 22, 23, 24, ...
get a(6) = 9 and remove the 9th term (= 21), etc.
- Reinhard Zumkeller, Feb 04 2014

Mathematica

Ceiling[Range[110]Sqrt[2]] (* Harvey P. Dale, Oct 30 2013 *)

Prog

(PARI) a(n)=ceil(n*sqrt(2)) \\ Charles R Greathouse IV, Oct 24 2011
(PARI) a(n)=sqrtint(2*n^2+sqrtint(8*n^2)+1) \\ Charles R Greathouse IV, Oct 24 2011
(Haskell)
a087057 n = a087057_list !! (n-1)
a087057_list = f [2..] where
   f (x:xs) = x : f (us ++ vs) where (us, _ : vs) = splitAt (x - 1) xs
-- Reinhard Zumkeller, Feb 04 2014

Crossrefs

Cf. A001951, A087055, A087056, A087058, A087059, A087060.

Sequence in context: A039033 A047333 A032796 * A184580 A184622 A195129

Adjacent sequences: A087054 A087055 A087056 * A087058 A087059 A087060

Keyword

easy,nonn

Author

Jens Voß, Aug 07 2003

Status

approved