cp's OEIS Frontend

A175035 Offsets i such that i + n*(n+1)/2 is a perfect square for some positive integer n.

%I A175035 #16 Sep 07 2019 15:51:54
%S A175035 1,3,4,6,8,9,10,13,15,16,19,21,22,24,25,26,28,30,33,34,35,36,39,43,45,
%T A175035 46,48,49,53,54,55,58,60,61,63,64,66,71,72,75,76,78,79,80,81,85,89,90,
%U A175035 91,93,94,97,99,100,103,105,106,108,111,114,115,116,118,120
%N A175035 Offsets i such that i + n*(n+1)/2 is a perfect square for some positive integer n.
%C A175035 The ansatz n*(n+1)/2+i=s^2 can be transformed into (2*n+1)^2-2*(2*s)^2 =1-8*i.
%C A175035 A necessary condition for solutions to this Diophantine equation is that D=2 is a quadratic residue of the squarefree part of 8*i-1 (see A057126).
%C A175035 A sufficient condition is then available by a sequence of tests on the continued fractions of a quadratic surd that originates from a solution of this congruence.
%C A175035 See Mollin and Matthews for details. - _R. J. Mathar_, Nov 16 2009
%H A175035 R. A. Mollin, <a href="http://dx.doi.org/10.1016/S0723-0869(01)80015-3">Simple continued fraction solutions for Diophantine Equations</a>, Exposit. Mathem. 19 (2001) 55-73.
%H A175035 Keith Matthews, <a href="http://www.numbertheory.org/pdfs/patz5.pdf">The Diophantine equation x^2-Dy^2=N,D >0</a>, Exposit. Mathem. 18 (4) (2000) 323-331 [<a href="http://www.ams.org/mathscinet-getitem?mr=1788328">MR1788328</a>].
%t A175035 Take[Rest[Ceiling[Sqrt[#]]^2-#&/@Accumulate[Range[1000]]//Union],70] (* _Harvey P. Dale_, Sep 07 2019 *)
%o A175035 (PARI) is(n)=#bnfisintnorm(bnfinit(z^2-8),-8*n+1) /* _Ralf Stephan_, Oct 14 2013 */
%Y A175035 Cf. A001108, A006451, A154138 to A154154.
%Y A175035 Cf. A230044.
%K A175035 easy,nonn
%O A175035 1,2
%A A175035 _Ctibor O. Zizka_, Nov 10 2009
%E A175035 Extended by _R. J. Mathar_, Nov 26 2009