A004202 Skip 1, take 1, skip 2, take 2, skip 3, take 3, etc.

2, 5, 6, 10, 11, 12, 17, 18, 19, 20, 26, 27, 28, 29, 30, 37, 38, 39, 40, 41, 42, 50, 51, 52, 53, 54, 55, 56, 65, 66, 67, 68, 69, 70, 71, 72, 82, 83, 84, 85, 86, 87, 88, 89, 90, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132

Offset: 1

Author

Alexander Stasinski

Keywords

• nonn
• tabl

Comments

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

a(A000217(n)) = A002378(n). [Reinhard Zumkeller, Feb 12 2011]

Complement of A004201. Upper s(n)-Wythoff sequence (as defined in A184117), for s(n)=A002024(n)=floor[1/2+sqrt(2n)]. I.e., A004202(n) = A002024(n) + A004201(n), with A004201(1)=1 and for n>1, A004201(n) = least positive integer not yet in (A004201(1..n-1) union A004202(1..n-1)). - M. F. Hasler (following observations from R. J. Mathar), Feb 13 2011

Positions of record values in A256188 that are greater than 1: A014132(n) = A256188(a(n)). - Reinhard Zumkeller, Mar 26 2015

Examples

			Interpretation as  Wythoff sequence (from _Clark Kimberling_):
s = (1,2,2,3,3,3,4,4,4,4...) = A002024 (n n's);
a = (1,3,4,7,8,9,13,14,...) = A004201 = least number > 0 not yet in a or b;
b = (2,5,6,10,11,12,17,18,...) = A004202 = a+s.
From _Michael Somos_, May 03 2019: (Start)
As a triangular array
  2;
  5,  6;
  10, 11, 12;
  17, 18, 19, 20;
(End)
		

Links

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

Crossrefs

Cf. A004201, A007606, A064801.

Cf. A014132, A256188, A127739.

Programs

Haskell

a004202 n = a004202_list !! (n-1)
a004202_list = skipTake 1 [1..] where
   skipTake k xs = take k (drop k xs) ++ skipTake (k + 1) (drop (2*k) xs)
-- Reinhard Zumkeller, Feb 12 2011

Mathematica

a = Table[n, {n, 1, 210} ]; b = {}; Do[a = Drop[a, {1, n} ]; b = Append[b, Take[a, {1, n} ]]; a = Drop[a, {1, n} ], {n, 1, 14} ]; Flatten[b]
a[ n_] := If[ n < 1, 0, With[{m = Round@Sqrt[2 n]}, n + m (m + 1)/2]]; (* Michael Somos, May 03 2019 *)
Take[#,(-Length[#])/2]&/@Module[{nn=20},TakeList[Range[ nn+nn^2],2*Range[ nn]]]//Flatten (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 13 2019 *)

PARI

A004202(n) = n+0+(n=(sqrtint(8*n-7)+1)\2)*(n+1)\2  \\ M. F. Hasler, Feb 13 2011

PARI

{a(n) = my(m); if( n<1, 0, m=round(sqrt(2*n)); n + m*(m+1)/2)}; /* Michael Somos, May 03 2019 */

Python

from math import isqrt, comb
def A004202(n): return n+comb((m:=isqrt(k:=n<<1))+(k-m*(m+1)>=1)+1,2) # Chai Wah Wu, Jun 19 2024

Formula

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

T(n, k) = n^2 + k, for n>=1, k>=1 as a triangular array. a(n) = n + A127739(n). - Michael Somos, May 03 2019