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%I #29 Sep 17 2022 16:23:43
%S 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,
%T 20,3,29,11,39,20,0,30,9,41,19,53,30,6,42,17,55,29,2,42,14,56,27,71,
%U 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
%N Distance between n^2 and the smallest triangular number >= n^2.
%C Smallest positive Z such that x^2 + x - 2n^2 - 2Z = 0 has a solution in integer x.
%C a(A077241(m)) = 2.
%C 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]
%H Harvey P. Dale, <a href="/A230038/b230038.txt">Table of n, a(n) for n = 1..1000</a>
%e The smallest triangular number >= 7^2 is 55 and 55-49=6, so a(7)=6.
%p 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
%t 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 *)
%o (PARI) a(n)=t=floor((sqrt(8*n^2)-1)/2)+1;t*(t+1)/2-n^2
%Y Cf. A064784.
%K nonn
%O 1,2
%A _Ralf Stephan_, Oct 08 2013
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