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

%I A004202 #43 Aug 01 2024 21:47:37
%S A004202 2,5,6,10,11,12,17,18,19,20,26,27,28,29,30,37,38,39,40,41,42,50,51,52,
%T A004202 53,54,55,56,65,66,67,68,69,70,71,72,82,83,84,85,86,87,88,89,90,101,
%U A004202 102,103,104,105,106,107,108,109,110,122,123,124,125,126,127,128,129,130,131,132
%N A004202 Skip 1, take 1, skip 2, take 2, skip 3, take 3, etc.
%C A004202 a(n) are the numbers satisfying m < sqrt(a(n)) < m + 0.5 for some integer m. - _Floor van Lamoen_, Jul 24 2001
%C A004202 a(A000217(n)) = A002378(n). [_Reinhard Zumkeller_, Feb 12 2011]
%C A004202 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
%C A004202 Positions of record values in A256188 that are greater than 1: A014132(n) = A256188(a(n)). - _Reinhard Zumkeller_, Mar 26 2015
%H A004202 Reinhard Zumkeller, <a href="/A004202/b004202.txt">Table of n, a(n) for n = 1..10000</a>
%F A004202 a(n) = n + A000217(A002024(n)). - _M. F. Hasler_, Feb 13 2011
%F A004202 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
%e A004202 Interpretation as  Wythoff sequence (from _Clark Kimberling_):
%e A004202 s = (1,2,2,3,3,3,4,4,4,4...) = A002024 (n n's);
%e A004202 a = (1,3,4,7,8,9,13,14,...) = A004201 = least number > 0 not yet in a or b;
%e A004202 b = (2,5,6,10,11,12,17,18,...) = A004202 = a+s.
%e A004202 From _Michael Somos_, May 03 2019: (Start)
%e A004202 As a triangular array
%e A004202   2;
%e A004202   5,  6;
%e A004202   10, 11, 12;
%e A004202   17, 18, 19, 20;
%e A004202 (End)
%t A004202 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]
%t A004202 a[ n_] := If[ n < 1, 0, With[{m = Round@Sqrt[2 n]}, n + m (m + 1)/2]]; (* _Michael Somos_, May 03 2019 *)
%t A004202 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 *)
%o A004202 (Haskell)
%o A004202 a004202 n = a004202_list !! (n-1)
%o A004202 a004202_list = skipTake 1 [1..] where
%o A004202    skipTake k xs = take k (drop k xs) ++ skipTake (k + 1) (drop (2*k) xs)
%o A004202 -- _Reinhard Zumkeller_, Feb 12 2011
%o A004202 (PARI) A004202(n) = n+0+(n=(sqrtint(8*n-7)+1)\2)*(n+1)\2  \\ _M. F. Hasler_, Feb 13 2011
%o A004202 (PARI) {a(n) = my(m); if( n<1, 0, m=round(sqrt(2*n)); n + m*(m+1)/2)}; /* _Michael Somos_, May 03 2019 */
%o A004202 (Python)
%o A004202 from math import isqrt, comb
%o A004202 def A004202(n): return n+comb((m:=isqrt(k:=n<<1))+(k-m*(m+1)>=1)+1,2) # _Chai Wah Wu_, Jun 19 2024
%Y A004202 Cf. A004201, A007606, A064801.
%Y A004202 Cf. A014132, A256188, A127739.
%K A004202 nonn,tabl
%O A004202 1,1
%A A004202 Alexander Stasinski