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
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
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
KEYWORD
nonn,tabl
AUTHOR
Alexander Stasinski
STATUS
approved