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

1,1

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

Links

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

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

Example

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)

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 *)

Prog

(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
(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

Crossrefs

Cf. A004201, A007606, A064801.

Cf. A014132, A256188, A127739.

Sequence in context: A244731 A307562 A109150 * A013647 A377167 A255068

Adjacent sequences: A004199 A004200 A004201 * A004203 A004204 A004205

Keyword

nonn,tabl

Author

Alexander Stasinski

Status

approved