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

Author

Alexander Stasinski

Keywords

• nonn
• nice

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

Crossrefs

Cf. A002024, A004202, A007606.

Programs

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

Mathematica

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

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

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