A230038 Distance between n^2 and the smallest triangular number >= n^2.

0, 2, 1, 5, 3, 0, 6, 2, 10, 5, 15, 9, 2, 14, 6, 20, 11, 1, 17, 6, 24, 12, 32, 19, 5, 27, 12, 36, 20, 3, 29, 11, 39, 20, 0, 30, 9, 41, 19, 53, 30, 6, 42, 17, 55, 29, 2, 42, 14, 56, 27, 71, 41, 10, 56, 24, 72, 39, 5, 55, 20, 72, 36, 90, 53, 15, 71, 32, 90, 50, 9, 69, 27, 89, 46, 2, 66, 21, 87, 41, 109, 62

Offset: 1

Author

Ralf Stephan, Oct 08 2013

Keywords

• nonn

Comments

Smallest positive Z such that x^2 + x - 2n^2 - 2Z = 0 has a solution in integer x.

a(A077241(m)) = 2.

Apparently, a(n) is triangular itself if n is of form (2k+1)*A001109(m), whenever k < A003499(m), or m > some small constant, k >= 0 (see A230060). [Comment improved by Nathaniel Johnston, Oct 08 2013]

Examples

			The smallest triangular number >= 7^2 is 55 and 55-49=6, so a(7)=6.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A064784.

Programs

Maple

a := proc(n) local t: t := ceil((sqrt(1 + 8*n^2) - 1)/2): return t*(t+1)/2 - n^2: end proc: seq(a(n),n=1..100); # Nathaniel Johnston, Oct 08 2013

Mathematica

Module[{nn=200,tr},tr=Accumulate[Range[nn]];Table[SelectFirst[tr,#>=n^2&]-n^2,{n,Floor[Sqrt[tr[[-1]]]]}]] (* Harvey P. Dale, Sep 17 2022 *)

PARI

a(n)=t=floor((sqrt(8*n^2)-1)/2)+1;t*(t+1)/2-n^2