OFFSET
0,5
COMMENTS
a(A002378(n)) = 0; a(n^2) = n.
Table A049581 T(n,k) = |n-k| read by sides of squares from T(1,n) to T(n,n), then from T(n,n) to T(n,1). - Boris Putievskiy, Jan 29 2013
LINKS
Harvey P. Dale, Table of n, a(n) for n = 0..1000
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
FORMULA
a(n) = A004738(n+1) - 1.
Let u(1)=1, u(n) = n - u(n-sqrtint(n)) (cf. A037458); then a(0)=0 and for n > 0 a(n) = 2*u(n) - n. - Benoit Cloitre, Dec 22 2002
a(0)=0 then a(n) = floor(sqrt(n)) - a(n - floor(sqrt(n))). - Benoit Cloitre, May 03 2004
a(n) = |A196199(n)|. a(n) = |n - t^2 - t|, where t = floor(sqrt(n)). - Boris Putievskiy, Jan 29 2013 [corrected by Ridouane Oudra, May 11 2019]
a(n) = A000194(n) - A053188(n) = t - |t^2 - n|, where t = floor(sqrt(n)+1/2). - Ridouane Oudra, May 11 2019
EXAMPLE
a(10) = |10 - 3*4| = 2.
From Boris Putievskiy, Jan 29 2013: (Start)
The start of the sequence as table:
0, 1, 2, 3, 4, 5, 6, 7, ...
1, 0, 1, 2, 3, 4, 5, 6, ...
2, 1, 0, 1, 2, 3, 4, 5, ...
3, 2, 1, 0, 1, 2, 3, 4, ...
4, 3, 2, 1, 0, 1, 2, 3, ...
5, 4, 3, 2, 1, 0, 1, 2, ...
6, 5, 4, 3, 2, 1, 0, 1, ...
...
The start of the sequence as triangle array read by rows:
0;
1, 0, 1;
2, 1, 0, 1, 2;
3, 2, 1, 0, 1, 2, 3;
4, 3, 2, 1, 0, 1, 2, 3, 4;
5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5;
6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6;
7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7;
...
Row number r contains 2*r-1 numbers: r-1, r-2, ..., 0, 1, 2, ..., r-1. (End)
MAPLE
MATHEMATICA
a[0] = 0; a[n_] := Floor[Sqrt[n]] - a[n - Floor[Sqrt[n]]]; Table[a[n], {n, 0, 103}] (* Jean-François Alcover, Dec 16 2011, after Benoit Cloitre *)
Join[{0}, Module[{nn=150, ptci}, ptci=Times@@@Partition[Range[nn/2+1], 2, 1]; Table[Abs[n-Nearest[ptci, n]], {n, nn}][[All, 1]]]] (* Harvey P. Dale, Aug 29 2020 *)
PROG
(PARI) a(n)=sqrtint(n)-a(n-sqrtint(n))
(PARI) apply( {A053615(n)=(t=sqrt(n)\/1)-abs(t^2-n)}, [0..99]) \\ M. F. Hasler, Feb 01 2025
(Python) A053615 = lambda n: (t := round(n**.5)) - abs(t**2 - n) \\ M. F. Hasler, Feb 01 2025
CROSSREFS
KEYWORD
easy,nice,nonn,changed
AUTHOR
Henry Bottomley, Mar 20 2000
STATUS
approved