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A175035
Offsets i such that i + n*(n+1)/2 is a perfect square for some positive integer n.
6
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, 46, 48, 49, 53, 54, 55, 58, 60, 61, 63, 64, 66, 71, 72, 75, 76, 78, 79, 80, 81, 85, 89, 90, 91, 93, 94, 97, 99, 100, 103, 105, 106, 108, 111, 114, 115, 116, 118, 120
OFFSET
1,2
COMMENTS
The ansatz n*(n+1)/2+i=s^2 can be transformed into (2*n+1)^2-2*(2*s)^2 =1-8*i.
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).
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.
See Mollin and Matthews for details. - R. J. Mathar, Nov 16 2009
LINKS
R. A. Mollin, Simple continued fraction solutions for Diophantine Equations, Exposit. Mathem. 19 (2001) 55-73.
Keith Matthews, The Diophantine equation x^2-Dy^2=N,D >0, Exposit. Mathem. 18 (4) (2000) 323-331 [MR1788328].
MATHEMATICA
Take[Rest[Ceiling[Sqrt[#]]^2-#&/@Accumulate[Range[1000]]//Union], 70] (* Harvey P. Dale, Sep 07 2019 *)
PROG
(PARI) is(n)=#bnfisintnorm(bnfinit(z^2-8), -8*n+1) /* Ralf Stephan, Oct 14 2013 */
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Ctibor O. Zizka, Nov 10 2009
EXTENSIONS
Extended by R. J. Mathar, Nov 26 2009
STATUS
approved