A004201 Accept one, reject one, accept two, reject two, ...

1, 3, 4, 7, 8, 9, 13, 14, 15, 16, 21, 22, 23, 24, 25, 31, 32, 33, 34, 35, 36, 43, 44, 45, 46, 47, 48, 49, 57, 58, 59, 60, 61, 62, 63, 64, 73, 74, 75, 76, 77, 78, 79, 80, 81, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 133, 134, 135

Offset

1,2

Comments

a(n) are the numbers satisfying m - 0.5 < sqrt(a(n)) <= m for some positive integer m. - Floor van Lamoen, Jul 24 2001

Lower s(n)-Wythoff sequence (as defined in A184117) associated to s(n) = A002024(n) = floor(1/2+sqrt(2n)), with complement (upper s(n)-Wythoff sequence) in A004202.

Links

Harvey P. Dale, Table of n, a(n) for n = 1..10000

Formula

a(n) = A061885(n-1)+1. - Franklin T. Adams-Watters, Jul 05 2009

a(n+1) - a(n) = A130296(n+1). - Reinhard Zumkeller, Jul 16 2008

a(A000217(n)) = n^2. - Reinhard Zumkeller, Feb 12 2011

a(n) = A004202(n)-A002024(n). - M. F. Hasler, Feb 13 2011

a(n) = n+A000217(A003056(n-1)) = n+A000217(A002024(n)-1). - M. F. Hasler, Feb 13 2011

a(n) = n + t(t+1)/2, where t = floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Dec 13 2012

a(n) = (2*n - r + r^2)/2, where r = round(sqrt(2*n)). - Wesley Ivan Hurt, Sep 20 2021

Mathematica

f[x_]:=Module[{c=1-x+x^2}, Range[c, c+x-1]]; Flatten[Array[f, 20]] (* Harvey P. Dale, Jul 31 2012 *)

Prog

(Haskell)
a004201 n = a004201_list !! (n-1)
a004201_list = f 1 [1..] where
   f k xs = us ++ f (k + 1) (drop (k) vs) where (us, vs) = splitAt k xs
-- Reinhard Zumkeller, Jun 20 2015, Feb 12 2011
(PARI) A004201(n)=n+(n=(sqrtint(8*n-7)+1)\2)*(n-1)\2  \\ M. F. Hasler, Feb 13 2011
(Python)
from math import comb, isqrt
def A004201(n): return n+comb((m:=isqrt(k:=n<<1))+(k>m*(m+1)), 2) # Chai Wah Wu, Nov 09 2024

Crossrefs

Cf. A002024, A004202, A007606.

Sequence in context: A226227 A319736 A100452 * A109054 A350690 A363751

Adjacent sequences: A004198 A004199 A004200 * A004202 A004203 A004204

Keyword

nonn,nice

Author

Alexander Stasinski

Status

approved